Volume Worksheets (Chapter 6)

Transcription

1 Volume Worksheets (Chapter 6) Name page contents: date AP Free Response Area Between Curves 3-5 Volume b Cross-section with Riemann Sums 6 Volume b Cross-section Homework 7-8 AP Free Response Volume b Cross-Section 9-10 More Volume b Cross-section Homework 11-1 An Wa You Spin It (MVP) Volume b Revolution About x- and -axes with Riemann Sums discs 15 Volume b Revolution About x- and -axes with Integrals 16 Volume b Revolution Homework Chapter 6. Revolution About Other Axes 19 Revolution About Other Axes Homework 0-1 Volume b Revolution with Midpoint Riemann Sums shells Shells Homework 3-4 AP Free Response Volume b Rotation 5-6 Densit

2 AP Free Response Area Between Curves Calculator 1 1.) Let f and g be the functions given b f ( x) sin x and 4 gx ( ) 4 x. Let R be the shaded region in the first quadrant enclosed b the -axis and the graphs of f and g, and let S be the shaded region in the first quadrant enclosed b the graphs of f and g, as shown in the figure. a.) Find the area of R. b.) Find the area of S. AP Free Response No calculator..) The function f is defined b a.) Find f '( x ). f ( x) 5 x for 5 x 5. b.) Write an equation for the line tangent to the graph of f at x = 3. f ( x) 5 x 3 c.) Let g be the function defined b gx ( ) x 7 3 x 5 Is g continuous at x = 3? Use the definition of continuit to explain our answer. d.) Find the value of 5 x 5 x dx 0. page

3 Volume b Cross-section with Riemann Sums 1.) Consider the region enclosed b the graphs of f ( x) x, = 0, and x = 9. a.) Fill in the table. x x b.) Use the values in the table above to draw rectangles on the graph with height and width x. c.) The right side of each slice, the side perpendicular to the x axis and the x- plane, is a square with side length as a function of x. Fill in the table below. x total area of right side x volume of prism d.) Is the total volume an overestimate or an underestimate for the actual volume? Explain using Calculus. We can use Calculus to get a more precise answer if we make the slices infinitel thin and add up an infinite number of them. The limits of integration are the input values from the first slice to the last slice. The integrand represents the area of a slice. e.) Write an integral to find the exact volume of the solid. integral: volume: page 3

4 .) If we make the cross-sections horizontal instead of vertical, do ou think the volume will be the same? Note: We are still using the region enclosed b the graphs of = x, = 0, and x = 9. a.) Fill in the table to find the side length of a square cross section b.) Use the values in the table above to draw rectangles on the graph with height x = f() and width. c.) The bottom side of each slice, the side perpendicular to the axis and the x- plane, is a square with side length x as a function of, x = f(). 0 1 total area of bottom side d.) Write an integral to find the exact volume of the solid. The limits of integration are the input values from the first slice to the last slice. integral: volume: volume of prism e.) Does the exact volume change when the cross-sections change from vertical to horizontal? Explain. page 4

5 3.) Find the area of each shape in terms of. Draw a picture to help ou visualize the answer. a.) a square with side b.) a half-square, cut diagonall, with diagonal c.) a half-square, cut diagonall, with leg d.) an equilateral triangle with side e.) a semicircle with diameter 4.) Complete the chart Function cross sections perpendicular to x-axis Interval Definite integral Volume round or truncate to 3 places after the decimal log x Semi-circle [ 1, 10 ] x Equilateral Triangle [ 0, 9 ] ln x Square [ 1, 6e ] 3 x Half-square Cut diagonall; (hpotenuse is in the x- plane) [ 1, ] page 5

6 Volume b Cross-section Homework For each problem, draw a figure and set up an integral. 1.) (calculator) Find the volume of the solid whose base is bounded b the graphs of x 1 and x 1, with the indicated cross sections taken perpendicular to the x-axis. a.) Squares b.) Rectangles of height.) (no calculator) Find the volume of the solid whose base is bounded b the circle with the indicated cross sections taken perpendicular to the x-axis. a.) Semicircles b.) Equilateral triangles x ) (no calculator) The base of a solid is bounded b x, 0, and x. Find the volume of the solid for each of the following cross sections taken perpendicular to the -axis. a.) Squares b.) Isosceles right triangles with a leg on the base of the solid page 6

7 AP Free Response Volume b Cross-section 1.) no calculator Let R be the region bounded b the graphs of x x and as shown in the figure to the right. The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are squares. Find the volume of this solid..) calculator allowed Let R be the region enclosed b the graph of 4 3 f ( x) x.3x 4 and the horizontal line 4, as shown in the figure to the right. Region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is an isosceles right triangle with a leg in R. Find the volume of the solid. page 7

8 3.) 1 Let f ( x) x 6x 4 and g( x) 4cos x. Let R be the region bounded b the graphs of f 4 and g, as shown in the figure above. The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Write, but do not evaluate, an integral expression that gives the volume of the solid. 4.) calculator allowed Let R be the region in the first quadrant bounded b the x-axis and the graphs of 5 x, as shown in the figure above. ln x and Region R is the base of a solid. For the solid, each cross section perpendicular to the x-axis is a square. Write, but do not evaluate, an expression involving one or more integrals that gives the volume of the solid. page 8

9 More Volume b Cross-section Homework 1.) Let R be the region in the first quadrant enclosed b the graphs of x and a.) Find the area of R. x. b.) Imagine a vertical line x k which divides the region R into two equal values. Find the value of k. c.) The region R is the base of a solid. For this solid, at each x-value the cross section perpendicular to the x-axis is an equilateral triangle. Find the volume of this solid. d.) Another solid has the same base R. For this solid, the cross sections perpendicular to the - axis are squares. Find the volume of this solid..) Let R be the region bounded b the graphs of sin( x) and x 3 4x. a.) Find the area of R. b.) The region R is the base of a solid. For this solid, each cross section perpendicular to the x- axis is a semi-circle. Find the volume of this solid. c.) The region R is the base of a solid. For this solid, each cross section perpendicular to the x- axis is an isosceles right triangle with the hpotenuse on the base of the solid. Find the volume of this solid. page 9

10 For questions 3-5, sketch a graph of the base. Draw and label a cross section of the solid. Find the volume of the solid. You should be able to evaluate these without a calculator. 3.) The base S is an elliptical region with boundar curve 9x Cross-sections perpendicular to the x-axis are isosceles right triangles with hpotenuse in the base. 4.) The base of S is the region enclosed b the parabola perpendicular to the -axis are squares. 1 x and the x-axis. Cross-sections 5.) The base of S is the region enclosed b the parabola 1 x and the x-axis. Cross-sections perpendicular to the x-axis are isosceles triangles with height equal to the base. page 10

11 An Wa You Spin It (MVP) Perhaps ou have used a potter wheel or a wood lathe. (A lathe is a machine that is used to shape a piece of wood b rotating it rapidl on its axis while a fixed tool is pressed against it. Table legs and wooden pedestals are carved on a wood lathe). You might have plaed with a spinning top or watched a figure skater spin so rapidl she looked like a solid blur. The cla bowl, the table leg, the rotating top and the spinning skater each of these can be modeled as solids of revolution a three dimensional object formed b spinning a two dimensional figure about an axis. Suppose the right triangle shown below is rotating rapidl about the x-axis. Like the spinning skater, a solid image would be formed b the blur of the rotating triangle. 1.) Draw and describe the solid of revolution formed b rotating this triangle about the x-axis..) Find the volume of the solid formed. 3.) What would this figure look like if the triangle rotates rapidl about the -axis? Draw and describe the solid of revolution formed b rotating this triangle about the -axis. 4.) Find the volume of the solid formed. page 11

12 5.) What about the following two-dimensional figure? Draw and describe the solid of revolution formed b rotating this figure about the x-axis. 6.) Draw a cross section of the solid of revolution formed b this figure if the plane cutting the solid is the plane containing the coordinate axes. 7.) Draw some cross sections of the solid of revolution formed b the figure if the planes cutting the solid are perpendicular to the x- axis and the plane containing the coordinate axes. Draw the cross sections when the intersecting planes are located at x = 5, x = 10, and x = ) How could ou use the cross-sections to estimate the volume? page 1

13 Volume b Revolution About x- and -axes with Riemann Sums - Discs 1.) = x a.) Fill in the table. x x b.) If we rotate the area under the curve in the first quadrant around the x-axis we form a solid. If we make vertical cuts perpendicular to the x-axis, the cross-sections will be circles. We will estimate the volume with four cuts, making four clinders. Each height,, is the radius of a circle and x is the width of the clinder. Use the values in the table above to draw four rectangles on the graph to represent the part of the solid that is in the x- plane. c.) The left side of each clinder, the side perpendicular to the x axis, is a circle with radius. Fill in the table below. x area of left side x 0 π8 18π 4 6 total volume of clinder We can use Calculus to get a more precise answer if we make the slices infinitel thin and add up an infinite number of them. The limits of integration are the input values from the first slice to the last slice. The integrand represents the area of a slice. d.) Write an integral to find the exact volume of the solid. integral: volume: page 13

14 .) If we rotate the area under the curve in the first quadrant around the -axis and make the cross-sections horizontal instead of vertical, do ou think the volume will be the same? = x a.) Fill in the table. x b.) Each height x f ( ), is the radius of a circle and is the width of a clinder. Use the values in the table above to draw rectangles on the graph to represent the part of the solid that is in the x- plane. c.) The bottom side of the clinder, the side perpendicular to the axis, is a circle with radius x f ( ). Fill in the table below. area of bottom side total d.) Write an integral to find the exact volume of the solid. The limits of integration are the input values from the first slice to the last slice. integral: volume: volume of clinder e.) Does the exact volume change when the cross-sections change from vertical to horizontal? Explain. page 14

15 Volume b Revolution About x- and -axes with Integrals 3.) Use Calculus to find the volume of this cone. 4.) Find the volume of the solid generated b rotating x about the x-axis from x 0 to x 1. 5.) Find the volume of the solid generated b 3 rotating x about the -axis from 0 to 8. 6.) Find the volume of the solid bounded b x and x rotated about the x-axis. page 15

16 Volume b Revolution Homework no calculator Find the volume of the solid obtained b rotating the region bounded b the given curves about the specified line. Sketch the region, the solid, and a tpical disc or washer. 1.) 1 x, 0 ; about the x-axis.) 5 x, 0, x, x 4 ; about the x-axis 3.) x, x 0 ; about the -axis 4.) 1 4 x, 5 x ; about the x-axis 5.) 1 4 x, x, 0 ; about the -axis page 16

17 Chapter 6. Revolution about Other Axes Let S be the region in the first Quadrant bounded b the graphs of the -axis. (calculator allowed) x, 1 0,5 x, and Example 1: Find the volume of the solid generated when S is revolved about the line. Example : Find the volume of the solid generated when S is revolved about the line 3. Example3: Find the volume of the solid generated when S is revolved about the line x 3. page 17

18 Example 4: Find the volume of the solid generated when S is rotated about the line x 1. Let Q be the region in the first quadrant between the -axis and the graph of ( 3) 9 x Example 5: Find the volume of the solid generated when Q is revolved about the line x 10.. page 18

19 Revolution About Other Axes Homework For questions 1 4, find the volume of the solid obtained b rotating the region bounded b the given curves about the specified line. Sketch the region, the solid, and a tpical disc or washer. Calculator allowed. 1.) x, x ; about 1.) 1 sec x on the interval [.5,.5], 3 ; about 1 3.) x, 1 x ; about x 1 4.) x, x ; about x 1 For questions 5-6, set up, but do not evaluate, an integral for the volume of the solid obtained b rotating the region bounded b the given curves about the specified line. 5.) 3 tan x, 1, x 0 ; about 1 6.) 0, sin x, 0 x ; about 1 page 19

20 Volume b Revolution with Midpoint Riemann Sums Shells 1.) = x a.) Fill in the table. x x b.) If we rotate the area under the curve in the first quadrant around the -axis we form a solid. We can slice the solid into vertical clinders (some hollow). Use the values in the table above to draw rectangles on the graph to represent the part of each of the four clinders that is in the x- plane. Color can be useful to show the different sections. clinder dimensions radius: x Height: c.) Fill in the table below. x lateral area rh x volume of clinder 1 π(1)(7.875) 31.5π total We can use Calculus to get a more precise answer if we make the slices infinitel thin and add up an infinite number of them. The limits of integration are the input values from the first slice to the last slice. The integrand represents the area of a slice. d.) Write an integral to find the exact volume of the solid. integral: volume: page 0

21 .) = x a) Fill in the table with positive values. x b.) If we rotate the area under the curve in the first quadrant around the x-axis we form a solid. We can slice the solid into horizontal clinders (some hollow). Use the values in the table above to draw rectangles on the graph to represent the part of each clinder that is in the x- plane. c.) hollow clinder dimensions radius: Height: x f ( ) d lateral area rh total volume clinder d.) Compare our estimate with our estimate from page 13 #1c. e.) Write an integral to find the exact volume of the solid. The limits of integration are the input values from the first slice to the last slice. integral: volume:. page 1

22 Shells Homework For question 1-3, use the method of clindrical shells to find the volume generated b rotating the region bounded b the given curves about the -axis. Sketch the region of a tpical shell. 1.) 1 x, 0, x 1, x.), 4, x 0 x, 0 x 3.) (calculator allowed) 4 x, x x 4 7 Use the method of clindrical shells to set up an integral to find the volume generated b rotating the region bounded b the given curves about the x-axis. Sketch the region of a tpical shell. (calculator) 4.) 3 x, 8, 0 x 5.) x 1, x 6.) If the region shown in the figure is rotated about the -axis to form a solid use the Midpoint Rule with n 5 to estimate the volume of the solid. page

23 AP Free Response Volume b Rotation 1.) calculator 4 3 Let R be the region enclosed b the graph of f ( x) x.3x 4 and the horizontal line 4, as shown in the figure to the left. Find the volume of the solid generated when R is rotated about the horizontal line..) no calculator Let R be the region bounded b the graphs of x and, x as shown in the figure to the left. Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the -axis. 3.) no calculator Let f(x) = x 6x + 4 and g(x) = 4cos ( 1 πx). Let R be 4 the region bounded b the graphs of f and g, as shown in the figure to the left. Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line = 4. page 3

24 4.) calculator Let R be the region in the first quadrant bounded b the x-axis and the graphs of ln x and 5 x, as shown in the figure to the left. Write but do not evaluate an integral expression for the volume of the solid generated when R is rotated about the vertical line x 1. 5.) The tide removes sand from Sand Point Beach at a rate modeled b the function R, given b 4 t Rt ( ) 5sin 5 A pumping station adds sand to the beach at a rate modeled b the function S, given b 15t St () 1 3t Both R(t)and S(t) have units of cubic ards per hour and t is measured in hours for 0 t 6. At time t = 0, the beach contains 500 cubic ards of sand. (calculator allowed) a.) How much sand will the tide remove from the beach during this 6-hour period? Indicate units of measure. b.) Write an expression for Yt (), the total number of cubic ards of sand on the beach at time t. c.) Find the rate at which the total amount of sand on the beach is changing at time t 4. d.) For 0 t 6, at what time t is the amount of sand on the beach a minimum? What is the minimum value? Justif our answers. page 4

25 Densit A.) What is Densit? In Phsics: Population: B.) Problem with Constant Densit C.) Problems with changing Densit (there are infinite examples here are just a few) 1.) Amount of Soot on a road. The Springfield chemical plant is putting soot onto a road that runs West to East from both sides of the plant. Let x be the distance from the plant in km. The densit of soot put onto the road, in mg per kilometer, each da, is given b d 500(1 x ). Write, and evaluate (with a calculator), a definite integral which stands for the total amount of soot put onto the road each da..) Number of tents in a campground. A campground is in the shape of a rectangle measuring 1000 ds b 600 ds with a river running along one side of the longer sides. At a distance of x ds from the river, the densit of tents per square d is given b d x on a Fourth of Jul weekend. Find the number of tents in the campground. 3.) Cit population: Circle Cit, a tpical metropolis, is ver densel populated near its center, and its population graduall thins out toward the cit limits. In fact, its population densit is 10000(3 r) people per square mile at a distance r from the center. What is the total population of the cit? page 5

26 Densit Homework (calculator allowed on all problems except #3) 1.) A cit in the shape of a circle of radius 10 miles is growing. Its population densit is a function of the distance from the center of the cit. At a distance of r miles from the cit 5000 center its population densit is d people per square mile. 1 r a.) Write a definite integral which stands for the number of people within miles of the cit center. Evaluate this definite integral. b.) Write a definite integral which stands for the number of people who live between 5 and 10 miles from the cit center. Evaluate this definite integral..) The densit of cars (in cars per mile) down a stretch of Route 3 (the highwa that connects Boston and Cape Cod) on a Frida afternoon at 5:00pm can be approximated b the function d 83 34cos(0.5 x) where x represents the distance in miles from the Boston. Write a definite integral which stands for the number of cars on the 10-mile stretch. Evaluate our integral. 3.) A state forest is in the shape of a rectangle measuring 8 miles b 0 miles, with a highwa running along one of the 0 mile sides. Deer in the forest are more numerous awa from the highwa. In fact, the densit of their population is expressed b the function D 0.5x 1 where D is the number of deer per square mile at distance x from the highwa. Write a definite integral which stands for the total number of deer in the forest. Evaluate our integral algebraicall. (no calculator) 4.) An hour after starting the Walk for Hunger, the number of walkers per mile a distance x 3 miles from the start of the walk was given (approximatel) b W 0.17x 3.1x 14. 8x. Assume all walkers are walking along a straight, thin path. a.) Write a definite integral which stands for the number of walkers between 3 and 4 miles from the start. Evaluate our definite integral. b.) Write a definite integral which stands for the number of walkers who are within 0-5 miles of the start. Evaluate our definite integral. page 6