Areas, volumes and arc lengths by integration
Use the definite integral to find the area under a curve and between two curves, a volume of revolution, an arc length and a polar area.
- UK: Year 13
- France: grade 12
- Germany: grade 12
- Iran: grade 12
- Afghanistan: grade 12
- Syria: grade 12
- Palestine: grade 12
- Ukraine: grade 11
- Russia: grade 11
Guide
The area under y = f(x) from a to b is the definite integral of f(x)dx between a and b: find the antiderivative at b and at a and subtract. Between two curves always integrate "top minus bottom".
Turn the region round the x-axis and the volume is V = π∫(f(x))2dx: square the function and keep the π. The arc length is L = ∫√(1 + (f′(x))2)dx.
In polar coordinates the area swept out by the radius r is A = 12∫r2dθ. Keep the half and the square.