Areas and volumes by integration · Grade 12

In Grade 12 you learn: areas between curves.

Integration measures more than the area under one curve. Subtracting one curve from another gives the area between them. Adding up thin discs gives the volume of a solid made by spinning a curve around an axis, like a bowl, a vase or a nose cone.

Key facts

• Area between two curves: ∫ (top − bottom) dx between the x-values where they meet.
• Area between a curve and the y-axis: ∫ x dy, with x written in terms of y.
• Region between two curves rotated about the x-axis: V = π∫ (y₁² − y₂²) dx (outer minus inner).
• Mean value of f from x = a to x = b: 1/(b − a) ∫ f(x) dx.