Volumes of Solids Packet

AP Calculus BC Name_____________________________________ Date_______________ Period _______

Volumes of Solids Packet For the shaded region R in each figure, find the following: REGION I a) The area of R b) The volume when R is rotated about the x-axis. c) The volume when R is rotated about the y-axis. d) The volume when R is rotated about the line x = -2. e) The volume when R is rotated about the line y = 6. f) The volume when R is the base of a solid and the cross-section of this solid are rectangles perpendicular to the x-axis with the base twice as long as the height. g) The arc length of y = 4x – x2 for the interval [0, 2]

y=x2

x=± y (2,4)

(0,0)

y=4x-x2

x = ± 4− y +2

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Volume Review Packet

AP Calculus BC

REGION 2

y=x-2 x=y+2

a) The area of R b) The volume when R is rotated about the y-axis. c) The volume when R is rotated about the line x = 4. d) The volume when R is rotated about the line x = -7. e) The volume when R is the base of a solid and the cross-section of this solid are semicircles perpendicular to the y-axis.

(4,2)

(1. –1)

x=y2 y=± x

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Volume Review Packet

REGION 3 a) The area of R b) The volume when R is rotated about the x-axis. c) The volume when R is rotated about the line y = 9. d) The volume when R is rotated about the line x = -3. e) The volume when R is the base of a solid and the cross-section of this solid are right isosceles triangles with one leg perpendicular to the x-axis. f) The arc length of y = (x + 1)2 for the interval [-1, 2]

AP Calculus BC

y=(x+1)2

(2, 9)

x = ± y −1

(-1, 0)

y=3x+3 y −3 x= 3

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