The base of S is a circular disk with radius r. Parallel crosssections perpendicular to

16 Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. Sketch the region, the solid, and a typical disk or washer. 1. y 2 2 1 2 x, y 0, x 1, x 2 2. y 1 2 x 2 , y 0 3. y sx 2 1 , y 0, x 5 4. y s25 2 x 2 , y 0, x 2, x 4 5. y x 3 , y x, x > 0 6. y 1 4 x 2 , y 5 2 x 2

16 Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. Sketch the region, the solid, and a typical disk or washer. 1. y 2 2 1 2 x, y 0, x 1, x 2 2. y 1 2 x 2 , y 0 3. y sx 2 1 , y 0, x 5 4. y s25 2 x 2 , y 0, x 2, x 4 5. y x 3 , y x, x > 0 6. y 1 4 x 2 , y 5 2 x 2

16 Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. Sketch the region, the solid, and a typical disk or washer. 1. y 2 2 1 2 x, y 0, x 1, x 2 2. y 1 2 x 2 , y 0 3. y sx 2 1 , y 0, x 5 4. y s25 2 x 2 , y 0, x 2, x 4 5. y x 3 , y x, x > 0 6. y 1 4 x 2 , y 5 2 x 2

16 Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. Sketch the region, the solid, and a typical disk or washer. 1. y 2 2 1 2 x, y 0, x 1, x 2 2. y 1 2 x 2 , y 0 3. y sx 2 1 , y 0, x 5 4. y s25 2 x 2 , y 0, x 2, x 4 5. y x 3 , y x, x > 0 6. y 1 4 x 2 , y 5 2 x 2

16 Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. Sketch the region, the solid, and a typical disk or washer. 1. y 2 2 1 2 x, y 0, x 1, x 2 2. y 1 2 x 2 , y 0 3. y sx 2 1 , y 0, x 5 4. y s25 2 x 2 , y 0, x 2, x 4 5. y x 3 , y x, x > 0 6. y 1 4 x 2 , y 5 2 x 2

16 Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. Sketch the region, the solid, and a typical disk or washer. 1. y 2 2 1 2 x, y 0, x 1, x 2 2. y 1 2 x 2 , y 0 3. y sx 2 1 , y 0, x 5 4. y s25 2 x 2 , y 0, x 2, x 4 5. y x 3 , y x, x > 0 6. y 1 4 x 2 , y 5 2 x 2

78 Here we rotate about the y-axis instead of the x-axis. Find the volume of the solid obtained by rotating the region bounded by the given curves about the y-axis. Sketch the region, the solid, and a typical disk. 7. x 2sy , x 0, y 9 8. y ln x, y 1, y 2, x 0

78 Here we rotate about the y-axis instead of the x-axis. Find the volume of the solid obtained by rotating the region bounded by the given curves about the y-axis. Sketch the region, the solid, and a typical disk. 7. x 2sy , x 0, y 9 8. y ln x, y 1, y 2, x 0

Volume of a pancreas A CAT scan of a human pancreas shows cross-sections spaced 1 cm apart. The pancreas is 12 cm long and the cross-sectional areas, in square centimeters, are 0, 7.7, 15.2, 18.0, 10.3, 10.8, 9.7, 8.7, 7.7, 5.5, 4.0, 2.7, and 0. Use the Midpoint Rule to estimate the volume of the pancreas

A log 10 m long is cut at 1-meter intervals and its crosssectional areas A (at a distance x from the end of the log) are listed in the table. Use the Midpoint Rule with n 5 to estimate the volume of the log. x smd A sm2 d x smd A sm2 d 0 0.68 6 0.53 1 0.65 7 0.55 2 0.64 8 0.52 3 0.61 9 0.50 4 0.58 10

(a) If the region shown in the figure is rotated about the x-axis to form a solid, use the Midpoint Rule with n 4 to estimate the volume of the solid. 0 4 4 2 86 10 2 y x (b) Estimate the volume if the region is rotated about the y-axis. Again use the Midpoint Rule with n 4.

Volume of a birds egg CAS (a) A model for the shape of a birds egg is obtained by rotating about the x-axis the region under the graph of fsxd sax 3 1 bx 2 1 cx 1 dds1 2 x 2 Use a CAS to find the volume of such an egg. (b) For a red-throated loon, a 20.06, b 0.04, c 0.1, and d 0.54. Graph f and find the volume of an egg of this species.

1315 Find the volume of the described solid S. 13. S is a right circular cone with height h and base radius r

1315 Find the volume of the described solid S. 14. The base of S is a circular disk with radius r. Parallel crosssections perpendicular to the base are squares.

1315 Find the volume of the described solid S. 15. The base of S is an elliptical region with boundary curve 9x 2 1 4y 2 36. Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.

The base of S is a circular disk with radius r. Parallel crosssections perpendicular to the base are isosceles triangles with height h and unequal side in the base. (a) Set up an integral for the volume of S. (b) By interpreting the integral as an area, find the volume of S.

Find the volume common to two spheres, each with radius r, if the center of each sphere lies on the surface of the other sphere

Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles