Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — area between two curves
Find the area enclosed between y=x and y=x2.
- 1Find where they meet: x=x2, so x(x−1)=0.The intersection points give the limits.
- 2x=0 and x=1.
- 3Test x=0.5: the line gives 0.5, the parabola 0.25, so the line is on top.A sketch settles this fastest.
- 4Area =∫01(x−x2)dxUpper minus lower.
- 5=[2x2−3x3]01=21−31
- 6=61Positive, as an area must be ✓
Example 2 — a volume of revolution
The region under y=x from x=0 to x=2 is rotated about the x-axis. Find the volume.
- 1V=π∫aby2dxπy2 is the area of each circular cross-section.
- 2y2=x2Square before integrating — this is the step most often skipped.
- 3V=π∫02x2dx
- 4=π[3x3]02=38π
- 5Check: this is a cone of radius 2 and height 2, and 31πr2h=31π(4)(2)=38π ✓An independent formula confirming the calculus.
Example 3 — reversing the chain rule
Find ∫(2x+1)5dx.
- 1Raise the power by one: (2x+1)6As with the ordinary power rule.
- 2Divide by the new power, 6.
- 3Also divide by the coefficient of x inside, which is 2.This compensates for the factor the chain rule would introduce.
- 4∫(2x+1)5dx=12(2x+1)6+C6×2=12.
- 5Check by differentiating: 126(2x+1)5×2=(2x+1)5 ✓Always verify — it takes seconds.