1. Find the lateral (side) surface area of the cone generated by revolving the line segment. Lateral surface area 1 ba se circumference slant height 2

Size: px
Start display at page:

Download "1. Find the lateral (side) surface area of the cone generated by revolving the line segment. Lateral surface area 1 ba se circumference slant height 2"

Transcription

1 s Section. Area of Surfaces of Revolution. Find the lateral (side) surface area of the cone generated by revolving the line segment y x, x, about the x-axis. Check your answer with the geometry formula Lateral surface area ba se circumference slant height. Find the lateral surface area of the cone generated by revolving the line segment y x, x, about the y-axis. Check your answer with the geometry formula Lateral surface area ba se circumference slant height. Find the lateral surface area of the cone frustum generated by revolving the line segment y x, x, about the x-axis. Check your answer with the geometry formula Frustum surface area r r slant height. Find the lateral surface area of the cone frustum generated by revolving the line segment y x, x, about the y-axis. Check your answer with the geometry formula Frustum surface area r r slant height. Find the area of the surface generated by y x, x, x axis 6. Find the area of the surface generated by y x x,. x., x axis 7. Find the area of the surface generated by y x, x, x axis 8. Find the area of the surface generated by x y y, y axis

2 . Find the area of the surface generated by x y y, y axis 8 y x, x ; y axis (Hint: Express. / evaluate the integral S y ds with appropriate limits.) ds in terms of, and. Did you know that if you can cut a spherical loaf of bread into slices of equal width, each slice will have the same amount of crust? To see why, suppose the semicircle y r x shown here is revolved about the x-axis to generate a sphere. Let AB be an arc of the semicircle that lies above an interval of length h on the x-axis. Show that the area swept out by AB does not depend on the location of the interval. (It does depend on the length of the interval.)

3 Section. Area of Surfaces of Revolution Exercise Find the lateral (side) surface area of the cone generated by revolving the line segment y x, x, about the x-axis. Check your answer with the geometry formula Lateral surface area ba se circumference slant height S b y a x x x ba se circumference r slant height Lateral surface area ba se circumference slant height

4 Find the lateral surface area of the cone generated by revolving the line segment y x, x, about the y-axis. Check your answer with the geometry formula Lateral surface area ba se circumference slant height x x y y x y x y d S x c y y y 8 ba se circumference 8 slant height Lateral surface area ba se circumference slant height 8 8

5 Find the lateral surface area of the cone frustum generated by revolving the line segment y x, x, about the x-axis. Check your answer with the geometry formula Frustum surface area r r slant height b S y a x x x x 6 r r slant height Frustum surface area r r slant height

6 Find the lateral surface area of the cone frustum generated by revolving the line segment y x, x, about the y-axis. Check your answer with the geometry formula Frustum surface area r r slant height y x y x x y S y y y y r r slant height Frustum surface area r r slant height

7 Find the area of the surface generated by x x x S x x x x 7 u / du 7 u / du y x, x, x axis u x du x du x x u x u / u / /

8 Find the area of the surface generated by y x, x, x axis / y x x x / x x x x x x S x x x x / u du x u u x du x u / u / /

9 Find the area of the surface generated by / y y y y y y y / y S y y / y / / y y d y / / x y y, y axis d y / / 8 / /

10 Find the area of the surface generated by y / y x y y, y axis 8 y y y y S y /8 y y /8 / u du /8 u / du /8 u y du y / /8 / /

11 / y x, x ; y axis (Hint: Express ds in terms of, and evaluate the integral S x ds with appropriate limits.) / x x x x ds x x x x x x x x S x ds x x u du x u u x du x x u u 8

12 Did you know that if you can cut a spherical loaf of bread into slices of equal width, each slice will have the same amount of crust? To see why, suppose the semicircle y r x shown here is revolved about the x-axis to generate a sphere. Let AB be an arc of the semicircle that lies above an interval of length h on the x-axis. Show that the area swept out by AB does not depend on the location of the interval. (It does depend on the length of the interval.) y r x x x r x r x x r x r r x r r x ah S r x r a r x ah r a ah rx a r a h a rh

Applications of Integration. Copyright Cengage Learning. All rights reserved.

Applications of Integration. Copyright Cengage Learning. All rights reserved. Applications of Integration Copyright Cengage Learning. All rights reserved. 1 Arc Length and Surfaces of Revolution Copyright Cengage Learning. All rights reserved. 2 Objectives Find the arc length of

More information

Volume by Slicing (Disks & Washers)

Volume by Slicing (Disks & Washers) Volume by Slicing Disks & Washers) SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should reference Chapter 6. of the recommended textbook or the equivalent chapter in

More information

12 m. 30 m. The Volume of a sphere is 36 cubic units. Find the length of the radius.

12 m. 30 m. The Volume of a sphere is 36 cubic units. Find the length of the radius. NAME DATE PER. REVIEW #18: SPHERES, COMPOSITE FIGURES, & CHANGING DIMENSIONS PART 1: SURFACE AREA & VOLUME OF SPHERES Find the measure(s) indicated. Answers to even numbered problems should be rounded

More information

Volume by Slicing (Disks & Washers)

Volume by Slicing (Disks & Washers) Volume by Slicing (Disks & Washers) SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should reference Chapter 6.2 of the recommended textbook (or the equivalent chapter

More information

AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION AREA OF A SURFACE OF REVOLUTION h cut r πr h A surface of revolution is formed when a curve is rotated about a line. Such a surface is the lateral boundar of a solid of revolution of the tpe discussed

More information

CHAPTER 6: APPLICATIONS OF INTEGRALS

CHAPTER 6: APPLICATIONS OF INTEGRALS (Exercises for Section 6.1: Area) E.6.1 CHAPTER 6: APPLICATIONS OF INTEGRALS SECTION 6.1: AREA 1) For parts a) and b) below, in the usual xy-plane i) Sketch the region R bounded by the graphs of the given

More information

Unit #13 : Integration to Find Areas and Volumes, Volumes of Revolution

Unit #13 : Integration to Find Areas and Volumes, Volumes of Revolution Unit #13 : Integration to Find Areas and Volumes, Volumes of Revolution Goals: Beabletoapplyaslicingapproachtoconstructintegralsforareasandvolumes. Be able to visualize surfaces generated by rotating functions

More information

AP Calculus. Slide 1 / 95. Slide 2 / 95. Slide 3 / 95. Applications of Definite Integrals

AP Calculus. Slide 1 / 95. Slide 2 / 95. Slide 3 / 95. Applications of Definite Integrals Slide 1 / 95 Slide 2 / 95 AP Calculus Applications of Definite Integrals 2015-11-23 www.njctl.org Table of Contents Slide 3 / 95 Particle Movement Area Between Curves Volume: Known Cross Sections Volume:

More information

2.2 Volumes of Solids of Revolution

2.2 Volumes of Solids of Revolution 2.2 Volumes of Solids of Revolution We know how to find volumes of well-established solids such as a cylinder or rectangular box. What happens when the volume can t be found quite as easily nice or when

More information

Chapter 1: Symmetry and Surface Area

Chapter 1: Symmetry and Surface Area Chapter 1: Symmetry and Surface Area Name: Section 1.1: Line Symmetry Line of symmetry(or reflection): divides a shape or design into two parts. Can be found using: A mirra Folding Counting on a grid Section

More information

If ( ) is approximated by a left sum using three inscribed rectangles of equal width on the x-axis, then the approximation is

If ( ) is approximated by a left sum using three inscribed rectangles of equal width on the x-axis, then the approximation is More Integration Page 1 Directions: Solve the following problems using the available space for scratchwork. Indicate your answers on the front page. Do not spend too much time on any one problem. Note:

More information

Chapter 8: Applications of Definite Integrals

Chapter 8: Applications of Definite Integrals Name: Date: Period: AP Calc AB Mr. Mellina Chapter 8: Applications of Definite Integrals v v Sections: 8.1 Integral as Net Change 8.2 Areas in the Plane v 8.3 Volumes HW Sets Set A (Section 8.1) Pages

More information

Surface Area of Circular Solids - Lesson 12-3

Surface Area of Circular Solids - Lesson 12-3 Surface Area of Circular Solids - Lesson 12-3 Today we talked about the surface area of circular solids. We started by defining spheres, hemispheres, cylinders, and cones: Baroody Page 1 of 10 Baroody

More information

OML Sample Problems 2017 Meet 7 EVENT 2: Geometry Surface Areas & Volumes of Solids

OML Sample Problems 2017 Meet 7 EVENT 2: Geometry Surface Areas & Volumes of Solids OML Sample Problems 2017 Meet 7 EVENT 2: Geometry Surface Areas & Volumes of Solids Include: Ratios and proportions Forms of Answers Note: Find exact answers (i.e. simplest pi and/or radical form) Sample

More information

Geometry Surface Area and Volume of Pyramids and Cones.

Geometry Surface Area and Volume of Pyramids and Cones. Geometry 11.6 Surface Area and Volume of Pyramids and Cones mbhaub@mpsaz.org 11.6 Essential Question How do you find the surface area and volume of a pyramid or a cone? Geometry 1.3 Surface Area of Pyramids

More information

Chapter 6 Some Applications of the Integral

Chapter 6 Some Applications of the Integral Chapter 6 Some Applications of the Integral More on Area More on Area Integrating the vertical separation gives Riemann Sums of the form More on Area Example Find the area A of the set shaded in Figure

More information

5/27/12. Objectives 7.1. Area of a Region Between Two Curves. Find the area of a region between two curves using integration.

5/27/12. Objectives 7.1. Area of a Region Between Two Curves. Find the area of a region between two curves using integration. Objectives 7.1 Find the area of a region between two curves using integration. Find the area of a region between intersecting curves using integration. Describe integration as an accumulation process.

More information

Applications of Integration. Copyright Cengage Learning. All rights reserved.

Applications of Integration. Copyright Cengage Learning. All rights reserved. Applications of Integration Copyright Cengage Learning. All rights reserved. Volume: The Disk Method Copyright Cengage Learning. All rights reserved. Objectives Find the volume of a solid of revolution

More information

In this chapter, we will investigate what have become the standard applications of the integral:

In this chapter, we will investigate what have become the standard applications of the integral: Chapter 8 Overview: Applications of Integrals Calculus, like most mathematical fields, began with trying to solve everyday problems. The theory and operations were formalized later. As early as 70 BC,

More information

Area and Volume. where x right and x left are written in terms of y.

Area and Volume. where x right and x left are written in terms of y. Area and Volume Area between two curves Sketch the region and determine the points of intersection. Draw a small strip either as dx or dy slicing. Use the following templates to set up a definite integral:

More information

STRAND E: Measurement. UNIT 13 Areas Student Text Contents. Section Squares, Rectangles and Triangles Area and Circumference of Circles

STRAND E: Measurement. UNIT 13 Areas Student Text Contents. Section Squares, Rectangles and Triangles Area and Circumference of Circles UNIT 13 Areas Student Text Contents STRAND E: Measurement Unit 13 Areas Student Text Contents Section 13.1 Squares, Rectangles and Triangles 13. Area and Circumference of Circles 13.3 Sector Areas and

More information

Aim: How do we find the volume of a figure with a given base? Get Ready: The region R is bounded by the curves. y = x 2 + 1

Aim: How do we find the volume of a figure with a given base? Get Ready: The region R is bounded by the curves. y = x 2 + 1 Get Ready: The region R is bounded by the curves y = x 2 + 1 y = x + 3. a. Find the area of region R. b. The region R is revolved around the horizontal line y = 1. Find the volume of the solid formed.

More information

Mr. Whelan Name: Block:

Mr. Whelan Name: Block: Mr. Whelan Name: Block: Geometry/Trig Unit 10 Area and Volume of Solids Notes Packet Day 1 Notes - Prisms Rectangular Prism: How do we find Total Area? Example 1 6cm Find the area of each face: Front:

More information

Geometry Chapter 11 Review. 1 Find the surface area and volume of the figure. Where necessary, express your answer in terms of.

Geometry Chapter 11 Review. 1 Find the surface area and volume of the figure. Where necessary, express your answer in terms of. Geometry hapter 11 Review Name: ate: 1 Find the surface area and volume of the figure. Where necessary, express your answer in terms of. 206 in. 2 ; 192 in. 3 208 in. 2 ; 192 in. 3 212 in. 2 ; 194 in.

More information

Volume of Cylinders. Volume of Cones. Example Find the volume of the cylinder. Round to the nearest tenth.

Volume of Cylinders. Volume of Cones. Example Find the volume of the cylinder. Round to the nearest tenth. Volume of Cylinders As with prisms, the area of the base of a cylinder tells the number of cubic units in one layer. The height tells how many layers there are in the cylinder. The volume V of a cylinder

More information

Design and Communication Graphics

Design and Communication Graphics An approach to teaching and learning Design and Communication Graphics Solids in Contact Syllabus Learning Outcomes: Construct views of up to three solids having curved surfaces and/or plane surfaces in

More information

Study Guide and Review

Study Guide and Review State whether each sentence is or false. If false, replace the underlined term to make a sentence. 1. Euclidean geometry deals with a system of points, great circles (lines), and spheres (planes). false,

More information

Grades 7 & 8, Math Circles 20/21/22 February, D Geometry

Grades 7 & 8, Math Circles 20/21/22 February, D Geometry Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing 2D Geometry Review Grades 7 & 8, Math Circles 20/21/22 February, 2018 3D Geometry Two-dimensional shapes

More information

SOLID SHAPES M.K. HOME TUITION. Mathematics Revision Guides. Level: GCSE Foundation Tier

SOLID SHAPES M.K. HOME TUITION. Mathematics Revision Guides. Level: GCSE Foundation Tier Mathematics Revision Guides Solid Shapes Page 1 of 15 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Foundation Tier SOLID SHAPES Version: 1. Date: 10-11-2015 Mathematics Revision Guides Solid

More information

Geometry 2 Final Review

Geometry 2 Final Review Name: Period: Date: Geometry 2 Final Review 1 Find x in ABC. 5 Find x in ABC. 2 Find x in STU. 6 Find cos A in ABC. 3 Find y in XYZ. 7 Find x to the nearest tenth. 4 Find x in HJK. 8 Find the angle of

More information

2012 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Group Members:

2012 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Group Members: 01 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Group Members: Reference Sheet Formulas and Facts You may need to use some of the following formulas and facts

More information

Chapter 14 Mensuration Surface Area and Volume Conversion of one solid into another solid Some time we have to melt one solid and convert it to another shap. For example we have to convert a metallic sphere

More information

Cutoff.Guru. Recruitment16.in. Recruitment16.in copyright Geometry and Mensuration. Some important mensuration formulas are:

Cutoff.Guru. Recruitment16.in. Recruitment16.in copyright Geometry and Mensuration. Some important mensuration formulas are: Geometry and Mensuration Mensuration: Mensuration is the branch of mathematics which deals with the study of Geometric shapes, Their area, Volume and different parameters in geometric objects. Some important

More information

Surface Area and Volume

Surface Area and Volume Surface Area and Volume Level 1 2 1. Calculate the surface area and volume of each shape. Use metres for all lengths. Write your answers to 4 decimal places: a) 0.8 m Surface Area: Volume: b) 1 m 0.2 m

More information

(1) Page #2 26 Even. (2) Page 596 #1 14. (3) Page #15 25 ; FF #26 and 28. (4) Page 603 #1 18. (5) Page #19 26

(1) Page #2 26 Even. (2) Page 596 #1 14. (3) Page #15 25 ; FF #26 and 28. (4) Page 603 #1 18. (5) Page #19 26 Geometry/Trigonometry Unit 10: Surface Area and Volume of Solids Notes Name: Date: Period: # (1) Page 590 591 #2 26 Even (2) Page 596 #1 14 (3) Page 596 597 #15 25 ; FF #26 and 28 (4) Page 603 #1 18 (5)

More information

5.2 Any Way You Spin It

5.2 Any Way You Spin It SECONDARY MATH III // MODULE 5 MODELING WITH GEOMETRY 5.2 Perhaps you have used a pottery wheel or a wood lathe. (A lathe is a machine that is used to shape a piece of wood by rotating it rapidly on its

More information

Review 1. Richard Koch. April 23, 2005

Review 1. Richard Koch. April 23, 2005 Review Richard Koch April 3, 5 Curves From the chapter on curves, you should know. the formula for arc length in section.;. the definition of T (s), κ(s), N(s), B(s) in section.4. 3. the fact that κ =

More information

FORMULAS to UNDERSTAND & MEMORIZE

FORMULAS to UNDERSTAND & MEMORIZE 1 of 6 FORMULAS to UNDERSTAND & MEMORIZE Now we come to the part where you need to just bear down and memorize. To make the process a bit simpler, I am providing all of the key info that they re going

More information

Geometry 1-1. Non-collinear Points not on the same line. Need at least 3 points to be non-collinear since two points are always collinear

Geometry 1-1. Non-collinear Points not on the same line. Need at least 3 points to be non-collinear since two points are always collinear Name Geometry 1-1 Undefined terms terms which cannot be defined only described. Point, line, plane Point a location in space Line a series of points that extends indefinitely in opposite directions. It

More information

Applications of Integration. Copyright Cengage Learning. All rights reserved.

Applications of Integration. Copyright Cengage Learning. All rights reserved. Applications of Integration Copyright Cengage Learning. All rights reserved. Volume: The Shell Method Copyright Cengage Learning. All rights reserved. Objectives Find the volume of a solid of revolution

More information

Area of Circle, Sector and Segment

Area of Circle, Sector and Segment 1 P a g e m a t h s c l a s s x 1. Find the circumference and area of a circle of radius 10.5 cm. 2. Find the area of a circle whose circumference is 52.8 cm. 3. Afield is in the form of a circle. The

More information

Example: The following is an example of a polyhedron. Fill the blanks with the appropriate answer. Vertices:

Example: The following is an example of a polyhedron. Fill the blanks with the appropriate answer. Vertices: 11.1: Space Figures and Cross Sections Polyhedron: solid that is bounded by polygons Faces: polygons that enclose a polyhedron Edge: line segment that faces meet and form Vertex: point or corner where

More information

LECTURE 3-1 AREA OF A REGION BOUNDED BY CURVES

LECTURE 3-1 AREA OF A REGION BOUNDED BY CURVES 7 CALCULUS II DR. YOU 98 LECTURE 3- AREA OF A REGION BOUNDED BY CURVES If y = f(x) and y = g(x) are continuous on an interval [a, b] and f(x) g(x) for all x in [a, b], then the area of the region between

More information

USING THE DEFINITE INTEGRAL

USING THE DEFINITE INTEGRAL Print this page Chapter Eight USING THE DEFINITE INTEGRAL 8.1 AREAS AND VOLUMES In Chapter 5, we calculated areas under graphs using definite integrals. We obtained the integral by slicing up the region,

More information

x + 2 = 0 or Our limits of integration will apparently be a = 2 and b = 4.

x + 2 = 0 or Our limits of integration will apparently be a = 2 and b = 4. QUIZ ON CHAPTER 6 - SOLUTIONS APPLICATIONS OF INTEGRALS; MATH 15 SPRING 17 KUNIYUKI 15 POINTS TOTAL, BUT 1 POINTS = 1% Note: The functions here are continuous on the intervals of interest. This guarantees

More information

AB Student Notes: Area and Volume

AB Student Notes: Area and Volume AB Student Notes: Area and Volume An area and volume problem has appeared on every one of the free response sections of the AP Calculus exam AB since year 1. They are straightforward and only occasionally

More information

MAT01B1: Surface Area of Solids of Revolution

MAT01B1: Surface Area of Solids of Revolution MAT01B1: Surface Area of Solids of Revolution Dr Craig 02 October 2018 My details: acraig@uj.ac.za Consulting hours: Monday 14h40 15h25 Thursday 11h20 12h55 Friday 11h20 12h55 Office C-Ring 508 https://andrewcraigmaths.wordpress.com/

More information

Calculus III. Math 233 Spring In-term exam April 11th. Suggested solutions

Calculus III. Math 233 Spring In-term exam April 11th. Suggested solutions Calculus III Math Spring 7 In-term exam April th. Suggested solutions This exam contains sixteen problems numbered through 6. Problems 5 are multiple choice problems, which each count 5% of your total

More information

VOLUME OF A REGION CALCULATOR EBOOK

VOLUME OF A REGION CALCULATOR EBOOK 19 March, 2018 VOLUME OF A REGION CALCULATOR EBOOK Document Filetype: PDF 390.92 KB 0 VOLUME OF A REGION CALCULATOR EBOOK How do you calculate volume. A solid of revolution is a solid formed by revolving

More information

CONES, PYRAMIDS AND SPHERES

CONES, PYRAMIDS AND SPHERES 9 10 YEARS The Improving Mathematics Education in Schools (TIMES) Project CONES, PYRAMIDS AND SPHERES MEASUREMENT AND GEOMETRY Module 12 A guide for teachers - Years 9 10 June 2011 Cones, Pyramids and

More information

Lecture 11 (Application of Integration) Areas between Curves Let and be continuous and on. Let s look at the region between and on.

Lecture 11 (Application of Integration) Areas between Curves Let and be continuous and on. Let s look at the region between and on. Lecture 11 (Application of Integration) Areas between Curves Let and be continuous and on. Let s look at the region between and on. Definition: The area of the region bounded by the curves and, and the

More information

Revolve Vertices. Axis of revolution. Angle of revolution. Edge sense. Vertex to be revolved. Figure 2-47: Revolve Vertices operation

Revolve Vertices. Axis of revolution. Angle of revolution. Edge sense. Vertex to be revolved. Figure 2-47: Revolve Vertices operation Revolve Vertices The Revolve Vertices operation (edge create revolve command) creates circular arc edges or helixes by revolving existing real and/or non-real vertices about a specified axis. The command

More information

Chapter 7 curve. 3. x=y-y 2, x=0, about the y axis. 6. y=x, y= x,about y=1

Chapter 7 curve. 3. x=y-y 2, x=0, about the y axis. 6. y=x, y= x,about y=1 Chapter 7 curve Find the volume of the solid obtained by rotating the region bounded by the given cures about the specified line. Sketch the region, the solid, and a typical disk or washer.. y-/, =, =;

More information

Grades 7 & 8, Math Circles 20/21/22 February, D Geometry Solutions

Grades 7 & 8, Math Circles 20/21/22 February, D Geometry Solutions Faculty of Mathematics Waterloo, Ontario NL 3G1 Centre for Education in Mathematics and Computing D Geometry Review Grades 7 & 8, Math Circles 0/1/ February, 018 3D Geometry Solutions Two-dimensional shapes

More information

Answers to Geometry Unit 5 Practice

Answers to Geometry Unit 5 Practice Lesson 0- Answers to Geometry Unit 5 Practice. a. Rectangle; Sample answer: It has four right angles. b. length: units; width: 9 units A 5 bh 7 units. 96 units. a. Parallelogram; Sample answer: Opposite

More information

V = 2πx(1 x) dx. x 2 dx. 3 x3 0

V = 2πx(1 x) dx. x 2 dx. 3 x3 0 Wednesday, September 3, 215 Page 462 Problem 1 Problem. Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the region (y = x, y =, x = 2)

More information

MATH 104 Sample problems for first exam - Fall MATH 104 First Midterm Exam - Fall (d) 256 3

MATH 104 Sample problems for first exam - Fall MATH 104 First Midterm Exam - Fall (d) 256 3 MATH 14 Sample problems for first exam - Fall 1 MATH 14 First Midterm Exam - Fall 1. Find the area between the graphs of y = 9 x and y = x + 1. (a) 4 (b) (c) (d) 5 (e) 4 (f) 81. A solid has as its base

More information

We have already studied equations of the line. There are several forms:

We have already studied equations of the line. There are several forms: Chapter 13-Coordinate Geometry extended. 13.1 Graphing equations We have already studied equations of the line. There are several forms: slope-intercept y = mx + b point-slope y - y1=m(x - x1) standard

More information

notes13.1inclass May 01, 2015

notes13.1inclass May 01, 2015 Chapter 13-Coordinate Geometry extended. 13.1 Graphing equations We have already studied equations of the line. There are several forms: slope-intercept y = mx + b point-slope y - y1=m(x - x1) standard

More information

Assignment Guide: Chapter 11 Geometry (L3)

Assignment Guide: Chapter 11 Geometry (L3) Assignment Guide: Chapter 11 Geometry (L3) (136) 11.1 Space Figures and Cross Sections Page 692-693 #7-23 odd, 35 (137) 11.2/11.4 Surface Areas and Volumes of Prisms Page 703-705 #1, 2, 7-9, 11-13, 25,

More information

Strategy. Using Strategy 1

Strategy. Using Strategy 1 Strategy Using Strategy 1 Scan Path / Strategy It is important to visualize the scan path you want for a feature before you begin taking points on your part. You want to try to place your points in a way

More information

Geometry: Semester 2 Practice Final Unofficial Worked Out Solutions by Earl Whitney

Geometry: Semester 2 Practice Final Unofficial Worked Out Solutions by Earl Whitney Geometry: Semester 2 Practice Final Unofficial Worked Out Solutions by Earl Whitney 1. Wrapping a string around a trash can measures the circumference of the trash can. Assuming the trash can is circular,

More information

Pages do a,c,e only (for questions that have parts)

Pages do a,c,e only (for questions that have parts) use their knowledge of rectangles, parallelograms and triangles to deduce formulae for the area of a parallelogram, and a triangle, from the formula for the area of a rectangle solve problems involving

More information

Volumes of Solids of Revolution Lecture #6 a

Volumes of Solids of Revolution Lecture #6 a Volumes of Solids of Revolution Lecture #6 a Sphereoid Parabaloid Hyperboloid Whateveroid Volumes Calculating 3-D Space an Object Occupies Take a cross-sectional slice. Compute the area of the slice. Multiply

More information

x=2 26. y 3x Use calculus to find the area of the triangle with the given vertices. y sin x cos 2x dx 31. y sx 2 x dx

x=2 26. y 3x Use calculus to find the area of the triangle with the given vertices. y sin x cos 2x dx 31. y sx 2 x dx 4 CHAPTER 6 APPLICATIONS OF INTEGRATION 6. EXERCISES 4 Find the area of the shaded region.. =5-. (4, 4) =. 4. = - = (_, ) = -4 =œ + = + =.,. sin,. cos, sin,, 4. cos, cos, 5., 6., 7.,, 4, 8., 8, 4 4, =_

More information

We have already studied equations of the line. There are several forms:

We have already studied equations of the line. There are several forms: Chapter 13-Coordinate Geometry extended. 13.1 Graphing equations We have already studied equations of the line. There are several forms: slope-intercept y = mx + b point-slope y - y1=m(x - x1) standard

More information

I IS II. = 2y"\ V= n{ay 2 l 3 -\y 2 )dy. Jo n [fy 5 ' 3 1

I IS II. = 2y\ V= n{ay 2 l 3 -\y 2 )dy. Jo n [fy 5 ' 3 1 r Exercises 5.2 Figure 530 (a) EXAMPLE'S The region in the first quadrant bounded by the graphs of y = i* and y = 2x is revolved about the y-axis. Find the volume of the resulting solid. SOLUTON The region

More information

f( x ), or a solution to the equation f( x) 0. You are already familiar with ways of solving

f( x ), or a solution to the equation f( x) 0. You are already familiar with ways of solving The Bisection Method and Newton s Method. If f( x ) a function, then a number r for which f( r) 0 is called a zero or a root of the function f( x ), or a solution to the equation f( x) 0. You are already

More information

EXERCISE NO:13.1. If cubes are joined end to end, the dimensions of the resulting cuboid will be 4 cm, 4cm, 8 cm. 2 lb bh lh.

EXERCISE NO:13.1. If cubes are joined end to end, the dimensions of the resulting cuboid will be 4 cm, 4cm, 8 cm. 2 lb bh lh. Class X - NCERT Maths EXERCISE NO:1.1 Question 1: cubes each of volume 64 cm are joined end to end. Find the surface area of the resulting cuboids. Solution 1: Given that, Volume of cubes = 64 cm (Edge)

More information

AP CALCULUS BC PACKET 2 FOR UNIT 4 SECTIONS 6.1 TO 6.3 PREWORK FOR UNIT 4 PT 2 HEIGHT UNDER A CURVE

AP CALCULUS BC PACKET 2 FOR UNIT 4 SECTIONS 6.1 TO 6.3 PREWORK FOR UNIT 4 PT 2 HEIGHT UNDER A CURVE AP CALCULUS BC PACKET FOR UNIT 4 SECTIONS 6. TO 6.3 PREWORK FOR UNIT 4 PT HEIGHT UNDER A CURVE Find an expression for the height of an vertical segment that can be drawn into the shaded region... = x =

More information

Volume and Surface Area Unit 28 Remember Volume of a solid figure is calculated in cubic units and measures three dimensions.

Volume and Surface Area Unit 28 Remember Volume of a solid figure is calculated in cubic units and measures three dimensions. Volume and Surface Area Unit 28 Remember Volume of a solid figure is calculated in cubic units and measures three dimensions. Surface Area is calculated in square units and measures two dimensions. Prisms

More information

AP Calculus. Areas and Volumes. Student Handout

AP Calculus. Areas and Volumes. Student Handout AP Calculus Areas and Volumes Student Handout 016-017 EDITION Use the following link or scan the QR code to complete the evaluation for the Study Session https://www.surveymonkey.com/r/s_sss Copyright

More information

Write down a formula for the surface area of a Prism and a Cylinder

Write down a formula for the surface area of a Prism and a Cylinder Write down a formula for the surface area of a Prism and a Cylinder Quiz Thursday Naming Figures Cross Sections Nets Lateral Area, Surface Area Prisms and cylinders have 2 congruent parallel bases. A lateral

More information

MATH 104 First Midterm Exam - Fall (d) A solid has as its base the region in the xy-plane the region between the curve y = 1 x2

MATH 104 First Midterm Exam - Fall (d) A solid has as its base the region in the xy-plane the region between the curve y = 1 x2 MATH 14 First Midterm Exam - Fall 214 1. Find the area between the graphs of y = x 2 + x + 5 and y = 2x 2 x. 1. Find the area between the graphs of y = x 2 + 4x + 6 and y = 2x 2 x. 1. Find the area between

More information

y = 4x + 2, 0 x 1 Name: Class: Date: 1 Find the area of the region that lies under the given curve:

y = 4x + 2, 0 x 1 Name: Class: Date: 1 Find the area of the region that lies under the given curve: Name: Class: Date: 1 Find the area of the region that lies under the given curve: y = 4x + 2, 0 x 1 Select the correct answer. The choices are rounded to the nearest thousandth. 8 Find the volume of the

More information

Geometry 10 and 11 Notes

Geometry 10 and 11 Notes Geometry 10 and 11 Notes Area and Volume Name Per Date 10.1 Area is the amount of space inside of a two dimensional object. When working with irregular shapes, we can find its area by breaking it up into

More information

A lg e b ra II. Trig o n o m e tric F u n c tio

A lg e b ra II. Trig o n o m e tric F u n c tio 1 A lg e b ra II Trig o n o m e tric F u n c tio 2015-12-17 www.njctl.org 2 Trig Functions click on the topic to go to that section Radians & Degrees & Co-terminal angles Arc Length & Area of a Sector

More information

Chapter 8 Quiz 1. Area. 1. Area 2. The perimeter of the rectangle is 126 m. 3. Area 4. Area

Chapter 8 Quiz 1. Area. 1. Area 2. The perimeter of the rectangle is 126 m. 3. Area 4. Area Chapter 8 Quiz 1 1. 2. The perimeter of the rectangle is 126 m. 11 cm 7 cm 5 cm 38 m 3. 4. 11 m 1 5 cm 1 13 m 12 m 15 m 12 cm 12 cm 16 cm 25 m 20 cm 20 cm 5. The area of the obtuse triangle is 176 ft 2.

More information

Surface Area of Pyramids and Cones

Surface Area of Pyramids and Cones Practice A Find the surface area of each pyramid. 1. 2. _ 3. 4. _ 5. 6. _ 7. 8. _ Find the surface area of each cone. Use 3.14 for. 9. 10. _ 11. 12. _ Practice B Find the surface area of each pyramid.

More information

5 Applications of Definite Integrals

5 Applications of Definite Integrals 5 Applications of Definite Integrals The previous chapter introduced the concepts of a definite integral as an area and as a limit of Riemann sums, demonstrated some of the properties of integrals, introduced

More information

3D Modeling. Visualization Chapter 4. Exercises

3D Modeling. Visualization Chapter 4. Exercises Three-dimensional (3D) modeling software is becoming more prevalent in the world of engineering design, thanks to faster computers and better software. Two-dimensional (2D) multiview drawings made using

More information

PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES

PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES UNIT 12 PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES (A) Main Concepts and Results Let a line l and a point P not lying on it be given. By using properties of a transversal and parallel lines,

More information

Aldine ISD Benchmark Targets /Geometry SUMMER 2004

Aldine ISD Benchmark Targets /Geometry SUMMER 2004 ASSURANCES: By the end of Geometry, the student will be able to: 1. Use properties of triangles and quadrilaterals to solve problems. 2. Identify, classify, and draw two and three-dimensional objects (prisms,

More information

Perimeter. Area. Surface Area. Volume. Circle (circumference) C = 2πr. Square. Rectangle. Triangle. Rectangle/Parallelogram A = bh

Perimeter. Area. Surface Area. Volume. Circle (circumference) C = 2πr. Square. Rectangle. Triangle. Rectangle/Parallelogram A = bh Perimeter Circle (circumference) C = 2πr Square P = 4s Rectangle P = 2b + 2h Area Circle A = πr Triangle A = bh Rectangle/Parallelogram A = bh Rhombus/Kite A = d d Trapezoid A = b + b h A area a apothem

More information

Area of Regular Polygons

Area of Regular Polygons Area of Regular Polygons Name:_ Find the area of each regular polygon. Leave your answer in simplest (radical) form. If your answer does not have a radical form, then round to the nearest tenth. 8 14.4

More information

AP Calculus AB Worksheet Areas, Volumes, and Arc Lengths

AP Calculus AB Worksheet Areas, Volumes, and Arc Lengths WorksheetAreasVolumesArcLengths.n 1 AP Calculus AB Worksheet Areas, Volumes, and Arc Lengths Areas To find the area etween the graph of f(x) and the x-axis from x = a to x = we first determine if the function

More information

Scheme of Work Form 4 (Scheme A)

Scheme of Work Form 4 (Scheme A) Scheme of Work Form 4 (Scheme A) Topic A revision of number work Directed Numbers Content Factors and Multiples Expressing a number as a product of prime factors LCM and HCF Objectives - Core and Paper

More information

In the first part of the lesson, students plot

In the first part of the lesson, students plot NATIONAL MATH + SCIENCE INITIATIVE Mathematics Using Linear Equations to Define Geometric Solids Level Geometry within a unit on volume applications Module/Connection to AP* Area and Volume *Advanced Placement

More information

KENDRIYA VIDYALAYA SANGATHAN, ERNAKULAM REGION. SECTION A (8 x 1 = 8)

KENDRIYA VIDYALAYA SANGATHAN, ERNAKULAM REGION. SECTION A (8 x 1 = 8) This question paper contains 06 pages KENDRIYA VIDYALAYA SANGATHAN, ERNAKULAM REGION Class: X SUMMATIVE ASSESSMENT - II Max. Marks : 90 Sub: Mathematics 2013-14 Time: 3 hrs General Instructions i) All

More information

Algebra II Trigonometric Functions

Algebra II Trigonometric Functions Slide 1 / 162 Slide 2 / 162 Algebra II Trigonometric Functions 2015-12-17 www.njctl.org Slide 3 / 162 Trig Functions click on the topic to go to that section Radians & Degrees & Co-terminal angles Arc

More information

11.4 Three-Dimensional Figures

11.4 Three-Dimensional Figures 11. Three-Dimensional Figures Essential Question What is the relationship between the numbers of vertices V, edges E, and faces F of a polyhedron? A polyhedron is a solid that is bounded by polygons, called

More information

Volumes of Rotation with Solids of Known Cross Sections

Volumes of Rotation with Solids of Known Cross Sections Volumes of Rotation with Solids of Known Cross Sections In this lesson we are going to learn how to find the volume of a solid which is swept out by a curve revolving about an ais. There are three main

More information

Implementation in COMSOL

Implementation in COMSOL Implementation in COMSOL The transient Navier-Stoke equation will be solved in COMSOL. A text (.txt) file needs to be created that contains the velocity at the inlet of the common carotid (calculated as

More information

B. Definition: The volume of a solid of known integrable cross-section area A(x) from x = a

B. Definition: The volume of a solid of known integrable cross-section area A(x) from x = a Mth 176 Clculus Sec. 6.: Volume I. Volume By Slicing A. Introduction We will e trying to find the volume of solid shped using the sum of cross section res times width. We will e driving towrd developing

More information

MATHEMATICS. Unit 1. Part 2 of 2. Expressions and Formulae

MATHEMATICS. Unit 1. Part 2 of 2. Expressions and Formulae MATHEMATICS Unit 1 Part 2 of 2 Expressions and Formulae Gradient Exercise 1 1) Work out the gradient of all the lines in the diagram. Write your answers in 1 y the form m AB T B 10 2 G H 8 6 4 F A C D

More information

To find the surface area of a pyramid and a cone

To find the surface area of a pyramid and a cone 11-3 Surface Areas of Pyramids and Cones Common Core State Standards G-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects. MP 1, MP 3, MP 4, MP 6, MP 7 Objective To find

More information

Math 10 C Measurement Unit

Math 10 C Measurement Unit Math 10 C Measurement Unit Name: Class: Date: ID: A Chapter Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which imperial unit is most appropriate

More information

Geometry Unit 10 Note Sheets Date Name of Lesson. 1.6 Two-Dimensional Figures Areas of Circles and Sectors

Geometry Unit 10 Note Sheets Date Name of Lesson. 1.6 Two-Dimensional Figures Areas of Circles and Sectors Date Name of Lesson 1.6 Two-Dimensional Figures 11.3 Areas of Circles and Sectors Quiz 11.1 Areas of Parallelograms and Triangles 11.2 Areas of Trapezoids, Rhombi and Kites 11.4 Areas of Regular Polygons

More information

Math 113 Exam 1 Practice

Math 113 Exam 1 Practice Math Exam Practice January 6, 00 Exam will cover sections 6.-6.5 and 7.-7.5 This sheet has three sections. The first section will remind you about techniques and formulas that you should know. The second

More information

Mensuration Formulas for SSC and Banking in PDF - Part 2

Mensuration Formulas for SSC and Banking in PDF - Part 2 Mensuration Formulas for SSC and Banking in PDF - Part 2 Mensuration is an important topic for Competitive Exam like SSC CGL, IBPS PO, SBI PO, IBPS Clerk, SBI Clerk, RBI Exams, Railway Exams, LIC AAO,

More information