Slice Solids into Disks

Every volume formula in this lesson is one idea: integrate the area of a representative slice.

Slicing Principle

Cut a solid perpendicular to an axis, and each slice has a measurable cross-sectional area. Volume is the integral of area times thickness. Nothing in this principle cares whether the solid is round; if you can write the slice area as a function, you can integrate it.

V = \int_a^b A(x)\,dx

Disks and Washers

Rotating a region around an axis sweeps out disks, each of area pi R squared. When the region leaves a hole around the axis, each slice is a washer, so subtract the inner radius: pi times R squared minus r squared. The rotation axis tells you whether the radius is measured vertically or horizontally.

V = \pi\int_a^b \bigl(R^2 - r^2\bigr)\,dx

A Worked Solid

Rotate the region under y = the square root of x from 0 to 1 around the x-axis. A slice at x is a disk of radius the square root of x, so its area is pi times x, and the integral gives pi over 2. Cylindrical shells offer the mirror method, integrating 2 pi x f(x) when slicing vertically around a horizontal axis is awkward.

V = \pi\int_0^1 x\,dx = \frac{\pi}{2}

Flashcard

V = \int A(x)\,dx

The general slicing method for volume

Cross-sectional area times an infinitesimal thickness is one slice's volume. Integrating along the axis sums every slice into the solid's volume.

Check your understanding

The region between y = x and y = 0 on [0, 2] is rotated about the x-axis. What setup gives the volume?

Take the unit quiz

← Previous lesson Next lesson →