Areas and volumes by integration

Integration finds areas between curves and volumes of solids made by spinning a curve around an axis.

Key facts

  • Area between curves: ∫ (top − bottom) dx between their crossing points.
  • Volume of revolution about the x-axis: V = π∫ y² dx.
  • About the y-axis: V = π∫ x² dy.

Worked examples

Example 1

Find the area between y = x and y = x².

  1. They meet at x = 0 and x = 1; between them y = x is on top.
  2. ∫_0^1 (x − x²) dx = [x²/2 − x³/3]_0^1 = 1/6

Answer: 1/6

Example 2

The curve y = √x from x = 0 to 4 is rotated about the x-axis. Find the volume.

  1. y² = x
  2. π∫_0^4 x dx = π[x²/2]_0^4 = 8π

Answer: 8π