What you'll learn
Using definite integrals to compute averages, areas, and volumes.
Average value
The average value of f on [a,b] is (1/(b−a)) ∫ₐᵇ f(x) dx.
Area between curves
∫ₐᵇ (top − bottom) dx (or right − left for dy). Find intersection points for the limits; split if the curves cross.
Accumulation
If a rate r(t) is given, total change = ∫ r(t) dt; final amount = initial + ∫ r(t) dt.
Volumes of revolution
- Disk method: V = π ∫ [R(x)]² dx.
- Washer method (with a hole): V = π ∫ ([R]² − [r]²) dx.
- Known cross-sections: V = ∫ A(x) dx (square, semicircle, etc.).
Exam tips
- Sketch the region; decide dx vs dy.
- For washers, subtract squares of radii, not the radii.
Common mistakes
- Forgetting the (b−a) divisor in average value.
- Squaring (R − r) instead of (R² − r²) for washers.