Key Idea: This topic finds areas against the y-axis and volumes of solids made by spinning a region 360°. On Paper 2 the GDC does the integral; on Paper 1 expect an exact answer.
Area between a curve and the y-axis
- Area = ∫ x dy with y-limits.
- Make x the subject first: x = g(y).
- Turn any x-limits into y-limits.
| Spin about | Volume | Radius of a slice |
|---|---|---|
| x-axis | V = π∫y² dx | y |
| y-axis | V = π∫x² dy | x |
| Between two curves | V = π∫(R² − r²) | R outer, r inner |
Square the whole function before you integrate.
Area between x = y² + 1, the y-axis, y = 0 and y = 2? 14/3.
y = √x from 0 to 4 spun about the x-axis? π∫₀⁴ x dx = 8π.
y = x + 1 from 0 to 2 spun about the x-axis? 26π/3.
Spinning about the y-axis: which limits? y-limits, with x² written in terms of y.