Area under a curve
If a curve y = f(x) lies above the x-axis between x = a and x = b, the area is ∫ab y dx. Measured from the y-axis instead, the area is ∫cd x dy.
The area between two curves (upper minus lower) is ∫ab (ytop − ybottom) dx.
Key formula
Volume of revolution about the x-axis: V = π ∫ab y² dx. About the y-axis: V = π ∫cd x² dy.
Worked example (area)
Worked example
Find the area under y = x² from x = 0 to x = 3. Area = ∫03 x² dx = [x³/3]03 = 27/3 − 0 = 9 units².
Worked example (volume)
The region under y = x from x = 0 to x = 2, rotated 360° about the x-axis, gives V = π ∫02 x² dx = π[x³/3]02 = 8π/3 units³.
Remember
- Square y before integrating for a volume about the x-axis.
- Area may be negative if the curve is below the axis — take the modulus.