08.08 Volumes with Cross Sections

The base of a solid is the region in the first quadrant bounded by the graphs of y = −x2 + 4, y = 6, and x = 2. For the solid, each cross-section perpendicular to the x-axis is a square. What is the volume of the solid?

25.067

The base of a solid is the region in the first quadrant between the graph of y = x2 and the x-axis for 0 ≤ x ≤ 2. For the solid, each cross-section perpendicular to the x-axis is a semicircle. What is the volume of the solid?

4 pi over 5

Let R be the region in the first quadrant bounded by the graphs of y = x2 and y = 2x. The region R is the base of a solid. For the solid, each cross-section perpendicular to the y-axis is a rectangle whose height is 4 times its width in the xy-plane. What is the integral expression that represents the volume of the solid?

integral from 0 to 4 of 4 times the quantity square root of y minus one half y end quantity squared dy

Let R be the region in the first quadrant bounded by the graph of y = 2x, the line x = 6, and the x-axis. R is the base of a solid whose cross sections perpendicular to the x-axis are equilateral triangles. What is the volume of the solid?

seventy two square root of 3

The base of a solid is a circle centered at the origin with a radius r, and every plane section perpendicular to a diameter is a square. What is the volume of the solid?

sixteen times r cubed over 3