Integration Techniques Lab
Beyond the power rule: u-substitution runs the chain rule backwards, integration by parts trades a hard integral for an easier one, and both unlock area between curves and volumes of revolution.
1 Area & volume explorer
Pick a pair of curves and drag the bounds. The canvas shades the region between them; toggle to volumes to spin the top curve around the x-axis into disks.
Setup: –
Value: –
2 What to notice
- The integrand is always top minus bottom — never the other way. If the curves cross inside [a, b], split the integral there.
- For volumes, the radius is squared: (x²)² = x⁴, not x². Forgetting the square is the classic error.
- Drag b past the crossing point and watch the “area” keep its sign discipline — net area, just like the FTC.