Key Idea: Integration finds areas of curved regions, and volumes of solids made by spinning a region around an axis. You set up the integral by hand, then let the GDC work it out.
| Region | Integral |
|---|---|
| Under a curve | A = ∫ₐᵇ y dx |
| Between two curves | A = ∫ₐᵇ (top − bottom) dx, limits at the intersections |
| Beside the y-axis | ∫ x dy, between two y-values |
Volumes of revolution
- About the x-axis: V = ∫ π y² dx.
- About the y-axis: V = ∫ π x² dy. Rewrite x² in terms of y and use y-limits.
- Each disc has area π·radius², so square the radius and keep the π.
- Between two curves: ∫ π(yₒᵤₜₑᵣ² − yᵢₙₙₑᵣ²) dx.
Spin a region and you get a stack of thin discs.
A panel lies between x = 4 − y² and the y-axis for 0 ≤ y ≤ 2. Area? ∫₀² (4 − y²) dy = 16/3 ≈ 5.33 m².
y = eˣ from 0 to 1 is spun about the x-axis. Volume? (π/2)(e² − 1) ≈ 10.0.
Between two curves, how do you know which is on top? Test one x-value in both.
A solid is spun about the y-axis. Which limits do you use? y-values, not x-values.