Volume of a Solid of Revolution

Solid of Revolution – Core Idea

• Formed by rotating a 2-D region under y=f(x) from x=a to x=b about the x-axis

• The swept region generates a 3-D object with rotational symmetry (e.g.
  vase, bell, lathe-turned part)

Cross-Section Analysis (Disk Method)

• Fix a specific x; consider the vertical segment of length f(x)

• When rotated, this segment becomes the radius of a circle

• Cross section ⟂ x-axis ⇒ disk of

  • Radius =f(x)

  • Area A(x)=\pi\,[f(x)]^2

Volume Formula

• Sum (integrate) all disk areas from a to b

• Volume of the solid:
  V=\int_a^b \pi\,[f(x)]^2\,dx

Key Takeaways

• Need same data as an area problem: limits a,b and function f(x)

• Method name: Disk Method (cross sections are solid disks)

• Applies to any shape with rotational symmetry about the axis of revolution