Determining Volumes by Slicing

Slicing a solid into known pieces

Washers and axes

Beyond circular sections

Choosing the method

A solid lying between x = a and x = b has cross-sectional area A(x) perpendicular to the x axis. What integral gives its volume?

The region under y = f(x) on [a, b] rotates about the x axis. What is the cross-sectional area A(x) of the resulting solid?

A washer has outer radius R and inner radius r. What is its cross-sectional area?

Rotating a region about the y axis, with slices perpendicular to the y axis, what plays the role of radius?

A solid with circular base radius r has cross sections perpendicular to a diameter that are squares. What is A(x)?

The semicircle y = sqrt(r^2 - x^2) rotated about the x axis gives a sphere. What volume integral computes it?

Rotating the line y = (r/h) x on [0, h] gives a cone of volume pi integral of (r^2/h^2) x^2 dx. As a fraction of pi r^2 h, what is the volume? Answer like 1/3.

The region between y = 2 - x^2 and y = x rotated about the x axis has outer radius 2 - x^2. What is the inner radius? Answer like x.

A volume integral multiplies area by thickness. If areas are in square centimeters and x in centimeters, the volume is in cubic centimeters. As a number, how many dimensions does the result measure? Answer with a number.

The thin slice thickness in volume integrals is written dx or dy. As a two-letter string, what is it when slicing perpendicular to the x axis? Answer in lowercase.