Volume of Solids of Revolution

• Disc method ($R$ is the radius of a representative disc)

  $$\textrm{Horizontal axis of revolution} V=\pi\int_a^b [R(x)]^2 dx$$

  $$\textrm{Vertical axis of revolution} V=\pi\int_c^d [R(y)]^2 dy$$

  
  

• Washer method ($R$ and $r$ are the outer and inner radii, respectively, of a representative washer)

  $$\textrm{Horizontal axis of revolution} V=\pi\int_a^b ([R(x)]^2-[r(x)]^2) dx$$

  $$\textrm{Vertical axis of revolution} V=\pi\int_c^d ([R(y)]^2-[r(y)]^2) dy$$

  
  

• Shell method ($p$ is the radius and $h$ is the height of a representative shell)

  $$\textrm{Horizontal axis of revolution} V=2\pi\int_c^d p(y)h(y) dy$$

  $$\textrm{Vertical axis of revolution} V=2\pi\int_a^b p(x)h(x) dx$$