Volumes with Cross Sections and by Rotation

The base of a solid is a circle bounded by the curve x²+y²=1. Find the volume of the solid formed by the cross sections perpendicular to the x-axis that are squares.

• $\frac{128}{3}$

• $\frac{64}{3}$

• $\frac{32}{3}$

• $\frac{16}{3}$

The base of a solid is a parabolic region bounded by the curves x²=9y and y = 1. Find the volume of the solid formed by the cross sections perpendicular to the y-axis that are rectangles with a height twice the length of the base.

• 36

• 16

• 8

• 64

The base of a solid is an ellipse bounded by the curve . Find the volume of the solid formed by 8x² + y² = 8 the cross sections perpendicular to the x-axis that are isosceles right triangles with a leg in the base.

• $\frac{32}{3}$

• $\frac{64}{3}$

• $\frac{128}{3}$

• $\frac{80\sqrt{2}}{3}$

Find the volume of the solid created by revolving the region bounded by the graph of y = x, the x-axis, and the line x = 2 about the line y = 0.

• $4\pi$

• $\frac{8}{3}\pi$

• $\frac{32}{5}\pi$

• $\frac{16}{3}\pi$

Find the volume of the region enclosed by f(x) = -x + 1, $x\ge0$  , and y = 0 rotated about the x-axis.

•  $\frac{\pi}{3}$  

•  $\frac{2\pi}{3}$  

•  $\frac{4\pi}{3}$  

•  $\frac{8\pi}{15}$  

The above graph gives a region shaded in the xy plane.

Find its volume when it is revolved around the y-axis.

• $2\pi$

• $4\pi$

• $6\pi$

• $8\pi$