Module 7: Applications of Integration
Calculus I – Study Guide
Learn Without Walls
1. Area Between Curves
A = ∫ab [f(x) − g(x)] dx (f on top)
A = ∫cd [f(y) − g(y)] dy (f on right)
Step 1: Find intersections. Step 2: Determine top/bottom (or right/left). Step 3: Integrate.
If curves cross, split the integral at each crossing point so the integrand stays non-negative.
2. Disk and Washer Methods
Slices are perpendicular to the axis of rotation.
Disk Method
V = π ∫ab [R(x)]² dx (about x-axis)
Use when: The region touches the axis of rotation (no gap).
Washer Method
V = π ∫ab {[R(x)]² − [r(x)]²} dx
Use when: There is a gap between the region and the axis (cross-section has a hole).
For non-standard axes (like y = k or x = k), measure R and r as distances from curves to that axis.
3. Shell Method
Slices are parallel to the axis of rotation. Each slice makes a cylindrical shell.
V = 2π ∫ab (radius)(height) dx
| Axis of Rotation | Radius | Height | Integrate |
|---|---|---|---|
| y-axis (x = 0) | x | f(x) | dx |
| x-axis (y = 0) | y | g(y) | dy |
| x = k | |x − k| | f(x) | dx |
Choose shells when disks/washers require solving for the other variable or produce multiple integrals.
4. Average Value of a Function
favg = (1/(b − a)) ∫ab f(x) dx
MVT for Integrals: If f is continuous on [a, b], there exists c in [a, b] with f(c) = favg.
Geometric meaning: favg is the height of the rectangle with the same area as the region under f on [a, b].