Applications of integration
The idea
Integration turns slicing into computation. Any quantity you can approximate by cutting a region or object into thin pieces — area between curves, volume of a solid, arc length, work done by a varying force, mass from a density — becomes a definite integral when the slices shrink. You already trust the area interpretation; applications generalize the same Riemann-sum logic to one new geometry at a time.
For solids of revolution, the disk method is the flagship: rotate the region under y = f(x) about the x-axis, and each vertical slice of thickness dx sweeps out a thin disk of radius f(x), hence volume π f(x)² dx. Summing slices gives V as the integral of π f(x)² from a to b. The reliable discipline is to draw one generic slice, write its tiny contribution, and only then integrate — the formula should be rebuilt, not recalled.
The standard trap is squaring carelessly: the disk integrand is π times the radius squared, and when there is a hole you need π(outer radius)² − π(inner radius)², which is not the same as π(outer − inner)². Squaring and subtracting do not commute, and the difference is exactly the cross term that the washer shape physically contains.
Worked example
The region under y = √x from x = 0 to x = 4 is rotated about the x-axis. Find the volume of the resulting solid.
- Picture one slice: at position x, a vertical strip of thickness dx sweeps out a disk of radius y = √x when rotated about the x-axis.
- Write the slice's volume: dV = π(radius)² dx = π(√x)² dx = πx dx — the square conveniently removes the root.
- Sum the slices as an integral: V equals the integral of πx from x = 0 to x = 4.
- Compute with the fundamental theorem: π × (x²/2 evaluated from 0 to 4) = π × (16/2 − 0) = 8π.
- Check against a bounding cylinder: the solid fits inside a cylinder of radius 2 and length 4 with volume 16π, and our 8π is exactly half of that — plausible for a shape that flares out like a horn from a point.
Answer. The solid of revolution has volume 8π ≈ 25.13 cubic units.
Check your understanding
- Why does the disk method square the function, and what geometric object does each slice represent?
- How would the setup change if the same region were rotated about the y-axis instead?
- Why is π(R² − r²) the right washer integrand rather than π(R − r)², and what does the difference represent?
- What other quantities besides volume can you compute by the slice-and-sum strategy, and what does a single slice contribute in each?
Build the foundations first
Applications of integration builds on these concepts. If any feel shaky, start there.