8.9 Volume with Disc Method: Revolving Around the x- or y-Axis

Learning objectives

—CHA-5.C—Calculate volumes of solids of revolution using definite integrals• CHA-5.C Calculate volumes of solids of revolution using definite integrals.• CHA-5.C.1 Volumes of solids of revolution around the x- or y-axis may be found by using definite integrals with the disc method.• Enduring understanding CHA-5: Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.

When a region touches the axis of rotation, a slice perpendicular to that axis sweeps out a solid circular disc. Its radius $R$ is the distance from the axis to the region's outer boundary, so a thin slice has volume approximately $\pi[R]^2$ times its thickness.

\text{about the }x\text{-axis: }V=\pi\int_a^b[R(x)]^2,dx,\qquad \text{about the }y\text{-axis: }V=\pi\int_c^d[R(y)]^2,dy

Draw slices perpendicular to the axis of rotation.
Use $dx$ for vertical discs about the $x$ -axis and $dy$ for horizontal discs about the $y$ -axis.
Express the radius as a nonnegative distance to the axis.
Integrate the disc area $\pi R^2$ over the stated bounds.

Example: rotate the region under $y=\sqrt{x}$ from $x=0$ to $x=4$ around the $x$ -axis. Each vertical slice forms a disc with $R(x)=\sqrt{x}$ . Therefore $V=\pi\int_0^4(\sqrt{x})^2\,dx=\pi\int_0^4x\,dx=\pi\left[\frac{x^2}{2}\right]_0^4=8\pi$ cubic units.

Square the radius, not the original function automatically: the radius must first be identified as a distance to the chosen axis. The disc formula also assumes no central hole; if the region does not reach the axis of rotation, subtracting an inner circular area is required instead.

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—8.10 Volume with Disc Method: Revolving Around Other AxesRead