Name: ______Period: ____
Calculus AB AP
Solids of Revolution
1. Let R be the region bounded by , , and the y-axis.
(a) Find the area of the region R.
(b) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Find the volume of this solid.
(c) A solid is generated by revolving the region R over the x-axis. Find the volume of this solid.
(d) A solid is generated by revolving the region R over the line . Find the volume of this solid.
2. Let R be the region in the first quadrant bounded by and .
(a) Find the area of the region R.
(b) Find the volume of the solid generated by revolving R over the y-axis.
(c) Find the volume of the solid generated by revolving R over the vertical line .
(d) Find the volume of the solid generated by revolving R over the horizontal line .
3. Let R be the region bounded by the graph of , the horizontal line , and the vertical line , as indicated in the figure.
(a) Find the area of the region R.
(b) Find the volume of the solid generated by revolving R over the y-axis.
(c) Find the volume of the solid generated by revolving R over the vertical line .
(d) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a semicircle. Find the volume of this solid.
4. Let R be the region bounded by the graph of , the horizontal line , and the vertical line .
(a) Find the area of the region R.
(b) Find the volume of the solid generated by revolving R over the x-axis.
(c) Find the volume of the solid generated by revolving R over the line .
(d) Find the volume of the solid generated by revolving R over the line .
5. Let , and .
(a) Find the region bounded by f and g.
(b) Let S be the region bounded by f and g. Find the volume of the solid generated by revolving S over the horizontal line .
(c) Let S be the region bounded by f and g. Find the volume of the solid generated by revolving S over the vertical line .
(d) The region bounded by f and g is the base of a solid. For this solid, each cross section perpendicular to the x-axis is an equilateral triangle. Find the volume of this solid.
6. Let , and . And let R and S be the regions bounded by f, g, and the axes, as indicated in the figure.
(a) Find the area of the region S.
(b) Find the volume of the solid generated by revolving R over the line .
(c) Find the volume of the solid generated by revolving S over the x-axis.
(d) Find the volume of the solid generated by revolving S over the y-axis.
7. Let , and . Let R be the region bounded by f , g, and the x-axis, as indicated in the figure.
(a) Find the area of the region R.
(b) Find the volume of the solid generated by revolving R over the y-axis.
(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 hypotenuse across the base. Find the volume of this solid.
(d) The region R is the base of a solid. For this solid, each cross section perpendicular to the y-axis is a rectangle with height six times the base. Find the volume of this solid.
Answer List:
1. Problem #1
(a) 1.997
(b) 2.347
(c) 30.271
(d) 7.375
2. Problem #2
(a) 1.599
(b) 8.779
(c) 11.310
(d) 25.061
3. Problem #3
(a) 1.056
(b) 13.900
(c) 6.006
(d) 0.218
4. Problem #4
(a) 6.638
(b) 77.337
(c) 35.629
(d) 89.495
5. Problem #5
(a) 2.395
(b) 20.987
(c) 34.032
(d) 1.584
6. Problem #6
(a) 0.744 *(Hint: Try splitting into two integrals.)
(b) 0.493
(c) 1.335 *(Hint: Try splitting into two integrals. Also try both disk and shell methods.)
(d) 3.252 *(Hint: Try splitting into two integrals. Also try both shell and washer methods.)
7. Problem #7
(a) 1.126 *(Hint: Try splitting into two integrals.)
(b) 16.131 *(Hint: Try splitting into two integrals. Also try both shell and washer methods.)
(c) 0.147 *(Hint: Try splitting into two integrals.)
(d) 12.945 *(Hint: Try solving for x in terms of y, and create an integral with respect to y.)
- 3 -