Volumes of Solids in Revolution

We can use definite integrals to find the volume of 3D solids by rotating curved around an axis to create a solid of revolution out of the region enclosed

Formulas

• Disk: $V=\pi {\int }_a^{b}R(x{)}^{2}dx$
• Washer: $V=\pi {\int }_a^b(R(x{)}^2-r(x{)}^2)dx$

Other axis of rotation

• Modify the formulas of R and r to be from the new axis
• Subtract it from each (the signs will work themselves out if the axis is negative)