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

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1 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 = 4 x *** (You will need to divide this into separate regions.) = x = x = x = x Prework for Part of Unit 4 continues on next page

2 AP CALCULUS BC PACKET FOR UNIT 4 SECTIONS 6. TO 6.3 Find the width of an horizontal segment that can be drawn into the shaded region x= 3 3 x = *** (You will need to divide this into separate regions.) x= x = x=

3 AP CALCULUS BC PACKET FOR UNIT 4 SECTIONS 6. TO 6.3 AREA BETWEEN CURVES (6.) The area A bounded b the curves = f ( x), g( x) continuous and f ( x) g( x) on [ ab, ], is: =, x= a, and x= b, where f and g are. Follow the steps to find the area enclosed b the curves = x, x 3 =, x =, and x =. a. Use a calculator to sketch and shade the region described. Determine an points of intersection. b. Use an integral or integrals to describe the area of the shaded region. c. Use our calculator to evaluate the integral(s) from part b. 3

4 AP CALCULUS BC PACKET FOR UNIT 4 SECTIONS 6. TO 6.3 Alternativel, the area A bounded b the curves x= f ( ), x g( ) f and g are continuous and f ( ) g( ) on [ ab, ], is =, = a, and = b, where. Follow the steps to find the area enclosed b the curves ( ) /5 above the x-axis. = x+ and x= + 3 a. Use a graphing utilit such as Desmos or Geogebra to sketch and shade the region described. Determine an points of intersection. b. Use an integral or integrals to describe the area of the shaded region. c. Use our calculator to evaluate the integral(s) from part b. 4

5 AP CALCULUS BC PACKET FOR UNIT 4 SECTIONS 6. TO 6.3 In general, imagine drawing an infinite number of rectangular slivers in the shaded region to cover the requested area. If ou draw vertical rectangles with thin bases, write our integral use Ytop Ybottom to find the height and dx as the width. If ou draw horizontal rectangles with thin heights, use Xright Xleft to find the base and d as the height. 3. Without a calculator, find the area enclosed b the curves f ( x) = sin x and g( x) = sin from x = to x = π. x 4. Use a calculator to find the area enclosed b the curves x 4 = and = 3x+. Answers:. ( 3 ) ( x x dx + x x ) dx. + 3 ( ) π 3. ( sin x + sin ) x dx ( ) d 3 4 d 5

6 AP CALCULUS BC PACKET FOR UNIT 4 SECTIONS 6. TO 6.3 Free online graphers for visualizing solids of revolution These links are available on the General Links page of the class website. Google lee ap calculus general links. Function Revolution: Graphs must be defined explicitl. Can revolve volume between two regions. Geogebra Volume of Solids of Revolution: Can onl graph one function revolved about x-axis. Shows a requested number of disk partitions along with integral expression and approximate volume. Geogebra Solid of Revolution: One function onl. Shows rotation about either axis. Can tilt view. 6

7 AP CALCULUS BC PACKET FOR UNIT 4 SECTIONS 6. TO 6.3 SOLIDS FORMED BY ROTATING ABOUT AN AXIS (6.) A surface of revolution is formed b revolving a two-dimensional curve about a line. The basic formula for volume is: Volume = (Area of Base) * (Height) For area between curves, we add areas of an infinite number of rectangles inside the region. For volume, imagine slicing a solid into an infinite number of disks. If we revolve a curve about the x-axis, then slicing perpendicular to the x-axis results in disks with cross-sectional area A( x ), where A is some function of x. We use dx to represent the thickness of each slice. The sum of volumes of slices from x=a to b x=b is represented b the integral A ( x ) dx. a Similarl, if we revolve a curve about the -axis, then slicing perpendicular to the -axis results in disks with crosssectional area A( ) and thickness d. A( ) dx d d The sum of volumes from =c to =d is A ( ) d. c Let s use this idea to derive the formula for the volume of a sphere.. First, we need to describe a sphere as a two-dimensional curve revolved about an axis. (a) Write the equation of a circle with center (,) and radius r. (b) Solve for in #a, disregarding the ±. (c) Sketch a graph of #b, and sketch a vertical rectangle anwhere inside the halfcircle. 7

8 AP CALCULUS BC PACKET FOR UNIT 4 SECTIONS 6. TO 6.3 (d) Imagine revolving the curve about the x-axis and slicing perpendicular to the x-axis to form disks. For the disk shown, describe the ends of the radius and the thickness in terms of x. (e) A(x) describes the area of the circular base of each disk. Write an expression for A(x). (f) Write a definite integral describing the volume of the sphere. (g) Evaluate and simplif.. Follow the steps to write an integral expression for the volume of the solid formed b 3 revolving the region bounded b = + x, =, x =, and x = about the x-axis: (a) Graph and shade the region. (b) Sketch a sample disk, and label the ends of its radius and thickness with appropriate expressions. (c) Write an expression for A(x), the cross-sectional area of the disk. (d) Write an integral expression for the volume. Do not evaluate. 8

9 AP CALCULUS BC PACKET FOR UNIT 4 SECTIONS 6. TO Follow the steps to find a formula for the volume obtained b rotating the region between = x, = ( x ) 3 +, x =, and x = about the x-axis. (a) Sketch and shade a D graph of the region. (You ma use a graphing utilit.) (b) To help ou picture this surface, experiment with one of the graphing utilities described on p. 6 of this packet. (c) Notice that slices here do not result in solid disks. Instead, we get washers. A side view and front view of a sample washer are shown below. R r Describe the cross-sectional area of a washer in terms of R, the radius of the outer circle, and r, the radius of the inner circle. (d) Since the graphs of x = x + intersect somewhere between x= and x=, we need to separate this volume into two different regions, one in which = x forms = and ( ) 3 the outside of the solid and the other in which ( ) 3 x-value do we need to separate the regions? = x + forms the outside. At what (e) Write variable expressions for R, r, and the thickness of the washer for each region. (f) Write a variable expression for A(x) in each region. (g) Write the volume of the solid as a definite integral. Do not evaluate. 9

10 AP CALCULUS BC PACKET FOR UNIT 4 SECTIONS 6. TO 6.3 SOLIDS FORMED BY ROTATING ABOUT A HORIZ/VERT LINE (6.) Complete these in our notebook. p. 39 # 7, 9,, 7: 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 disk or washer. 7. = x, = x; about the x-axis 9. = x, x= ; about the -axis. = x, = x ; about = 7. = x, x = ; about x = p. 39 #4: Use a computer algebra sstem to find the exact volume of the solid obtained b rotating the region bounded b the given curves about the specified line. π =, = xcos x 4 ; about = x x p. 39 #4, 43: Each integral represents the volume of a solid. Describe the solid. 4. π π ( ) d d p. 39 #45: A CAT scan produces equall spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained onl b surger. Suppose that a CAT scan of a human liver shows cross-sections spaced.5 cm apart. The liver is 5 cm long and the cross-sectional areas, in square centimeters, are, 8, 58, 79, 94, 6, 7, 8, 63, 39, and. Use the Midpoint Rule to estimate the volume of the liver.

11 AP CALCULUS BC PACKET FOR UNIT 4 SECTIONS 6. TO 6.3 VOLUME BY CROSS-SECTIONS (6.) p. 39 #55, 56, 59: Find the volume of the described solid S. 55. The base of S is an elliptical region with boundar curve 9x + 4 = 36. Cross-sections perpendicular to the x-axis are isosceles right triangles with hpotenuse in the base. 56. The base of S is the parabolic region ( ) equilateral triangles. { x, x }. Cross-sections perpendicular to the -axis are 59. The base of S is the triangular region with vertices (,), (3,), and (,). Cross-sections perpendicular to the -axis are isosceles triangles with height equal to the base.

12 AP CALCULUS BC PACKET FOR UNIT 4 SECTIONS 6. TO 6.3 VOLUMES BY CYLINDRICAL SHELLS (6.3) Complete these in our notebook. p. 396 #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 and a tpical shell. =, =, x =, x = x p. 396 #3: Use the method of clindrical shells to find the volume generated b rotating the region bounded b the given curves about the x-axis. Sketch the region and a tpical shell. = 4x, x+ = 6 p. 397 #3, 5: 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 axis. 3. π = x ; about x = 4 = x, sin 5. x= sin, π, x = ; about = 4 p. 397 #9, 3: Each integral represents the volume of a solid. Describe the solid π x 5 dx 3. π ( )( ) 3 d

13 AP CALCULUS BC PACKET FOR UNIT 4 SECTIONS 6. TO 6.3 PRACTICE VOLUME Find the volume of the solid of revolution formed when the region R is revolved around the given line. Set up the graph and integral without a calculator, and then use a calculator to evaluate the integral.. R: bounded between x x x =, = 3, =, and = ; about the x-axis. R: bounded between π x =, 4 π x =, = sin x, and = ; about the x-axis 3. R: bounded between x =, 5 x =, 4. R: bounded between = x+ 5 and = x, and = ; about the line = = x + 3; about the line = 5. R: bounded between = x and = x ; about the -axis 6. R: bounded between x 6 7. R: bounded between 6 x = and ( x 3) = and ( x ) = ; about the line x = = + ; about the line = 8. R: below = x, above the x-axis, and between x = and x = ; about the -axis 9. R: bounded between = x and a. Using disks/washers b. Using clindrical shells = x ; about the line x =. R: below = cos x, above the x-axis, and between x = and a. Using disks/washers b. Using clindrical shells π x = ; about the -axis. Find the volume of the solid whose base is the region bounded b are a. semicircles perpendicular to the -axis b. semicircles perpendicular to the x-axis c. equilateral triangles perpendicular to the -axis = x and = with cross-sections that. Find the volume of the solid whose base is a circular disk with radius r and cross-sections perpendicular to the x-axis are squares. 3

14 AP CALCULUS BC PACKET FOR UNIT 4 SECTIONS 6. TO 6.3 PRACTICE VOLUME SOLUTIONS 3. π ( x ) dx π π / π /4 3. ( x ) sin x dx.4 ( ) 5/ π dx 5.83 ( ) 4. ( ) ( ) π x + 5 x + 3 dx ( ) ( ) π d π ( ) d π ( ) ( ) ( ) 6 x x dx ( ) 8. ( ) π d.57 ( ) b. ( )( ) 9. a. ( ) ( ) π + + d.68 cos d π / b. π x( ). a. π ( ) π x + x x dx.68 cos x dx π d.785 b.. a. ( ) π.9 x dx c. 3 d.866 r. 4 ( ) r 6 r x dx = r 3 3 4

15 AP CALCULUS BC PACKET FOR UNIT 4 SECTIONS 6. TO 6.3 Old AP Exam Questions Area and Volume 3, # - Calculator Let R be the shaded region bounded b the graphs of = x and = e 3x and the vertical line x =, as shown in the figure above. (a) Find the area of R. (b) Find the volume of the solid generated when R is revolved about the horizontal line =. (c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a rectangle whose height is 5 times the length of its base in region R. Find the volume of this solid. 4, # Calculator Let f and g be the functions given b f ( x) = x( x) and ( ) 3( ) and g are shown in the figure above. g x = x x for x. The graphs of f (a) Find the area of the shaded region enclosed b the graphs of f and g. (b) Find the volume of the solid generated when the shaded region enclosed b the graphs of f and g is revolved about the horizontal line =. (c) Let h be the function given b h( x) = kx( x) for x. For each k >, the region (not shown) enclosed b the graphs of h and g is the base of a solid with square cross sections perpendicular to the x- axis. There is a value of k for which the volume of this solid is 5. Write, but do not solve, an equation involving an integral expression that could be used to find the value of k. 5

16 5, Form B - #6, NO Calculator AP CALCULUS BC PACKET FOR UNIT 4 SECTIONS 6. TO 6.3 = for x, as shown in the figure above. Let R be x + the region bounded b the graph of f, the x- and -axes, and the vertical line x= k, where k. Consider the graph of the function f given b f ( x) (a) Find the area of R in terms of k. (b) Find the volume of the solid generated when R is revolved about the x-axis in terms of k. (c) Let S be the unbounded region in the first quadrant to the right of the vertical line x= k and below the graph of f, as shown in the figure above. Find all values of k such that the volume of the solid generated when S is revolved about the x-axis is equal to the volume of the solid found in part (b)., #4 NO Calculator Let R be the region in the first quadrant bounded b the graph of = x, the horizontal line = 6, and the - axis, as shown in the figure above. (a) Find the area of R. (b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line = 7. (c) Region R is the base of a solid. For each, where 6, the cross section of the solid taken perpendicular to the -axis is a rectangle whose height is 3 times the length of its base in region R. Write, but do not evaluate, an integral expression that gives the volume of the solid. 6