How to find volume with the disk method

When a region touches its axis of rotation, a slice perpendicular to that axis can sweep out a solid disk. Its radius is the distance from the axis to the outer edge of the region. Once we know that radius, we can add up the cross-sectional areas with an integral.

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Find the disk area from the radius

$A=\pi R^2$

The disk fills the circle.

Rotate around the x-axis

Rotate the region between $y=2x^3$ and the $x$-axis over $1\le x\le 3$. The vertical slice has radius $2x^3$, and its thickness is measured along $x$.

$\begin{array}{l}V=\pi {\int }_1^3(2x^3{)}^{2}dx\\ =\frac{4\pi }{7}(3^7-1)=\frac{8744\pi }{7}\end{array}$

Rotate around the y-axis

Rotate the region between $x=3y^{-2}$ and the $y$-axis from $y=3$ to $y=5$. The disks come from horizontal slices, so the thickness is measured along $y$.

$\begin{array}{l}V=9\pi {\int }_3^5{y}^{-4}dy\\ =3\pi (3^{-3}-5^{-3})=\frac{98\pi }{1125}\end{array}$

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