Washer Method – Volume of Solids of Revolution

Solids of Revolution – Washer Method

• Region is bounded by two curves and revolved about an axis ⇒ each slice becomes a washer (disk with a hole).
• Outer curve ⟹ outer radius R(x); inner curve ⟹ inner radius r(x).
• Area of one washer: A(x)=\pi R(x)^2-\pi r(x)^2=\pi\big(R(x)^2-r(x)^2\big).
• Volume of the solid (axis of rotation parallel to x-axis):
  V=\int_{a}^{b} \pi \big(R(x)^2-r(x)^2\big)\,dx
• If the region is revolved about the y-axis (or any line parallel to it), use y-slices:
  V=\int_{c}^{d} \pi \big(R(y)^2-r(y)^2\big)\,dy
• Special case (solid disk): r(x)=0, so V=\int_{a}^{b} \pi R(x)^2\,dx.