Summary Notes: Volume of Revolution

VOLUME OF REVOLUTION

REVOLUTION ABOUT THE \(x\)-AXIS

\(V = \pi \displaystyle\int_{a}^{b} [\mathrm{f}(x)]^2 \, \mathrm{d}x = \pi \displaystyle\int_{a}^{b} y^2 \, \mathrm{d}x\)

REVOLUTION ABOUT THE \(y\)-AXIS

\(V = \pi \displaystyle\int_{a}^{b} [\mathrm{f}(y)]^2 \, \mathrm{d}y = \pi \displaystyle\int_{a}^{b} x^2 \, \mathrm{d}y\)

VOLUME BOUNDED BY TWO CURVES

REVOLUTION ABOUT THE \(x\)-AXIS

\(V = \pi \displaystyle\int_{a}^{b} [\mathrm{g}(x)]^2 - [\mathrm{f}(x)]^2 \, \mathrm{d}x = \pi \displaystyle\int_{a}^{b} (y_2)^2 - (y_1)^2 \, \mathrm{d}x\)

REVOLUTION ABOUT THE \(y\)-AXIS

\(V = \pi r^2 (c-a) - \displaystyle\int_{a}^{b} [\mathrm{f}(y)]^2 \, \mathrm{d}y - \displaystyle\int_{b}^{c} [\mathrm{g}(y)]^2 \, \mathrm{d}y\)

\(= \pi r^2 (c-a) - \displaystyle\int_{a}^{b} (x_1)^2 \, \mathrm{d}y - \displaystyle\int_{b}^{c} (x_2)^2 \, \mathrm{d}y\)