# Volume of revolution

Spin a region around the x-axis. Slice it into thin discs of radius f(x), each with volume pi f(x)^2 dx, and add them up with an integral.

Maths (university) · Part 06: Integration techniques and uses · Card 6 of 9 · Last updated 2026-10-19

## Analogy

A potter's wheel: spinning a flat shape gives a solid vase.

## Example

**y = sqrt(x), x from 0 to 4**

Radius is sqrt(x), so area = pi x.
Volume = integral of pi x dx from 0 to 4.
= pi [x^2/2] = pi (16/2).
Volume = 8 pi, about 25.1.

## In practice

Gives volumes of cones, spheres, bottles and tanks.

## Practice

### Flashcards

- **What is the disc volume formula?** V = pi times the integral of f(x)^2 dx.

## Questions people ask

### Why is the disc formula pi times f squared?

Each thin slice is a circle (disc) of radius f(x), and the area of a circle is pi times radius squared. Multiply by the slice thickness dx to get a volume. The integral adds up all slices from the start to the end.

### What if the region has a hole in the middle?

Use washers: discs with a hole. The area of each slice is pi times outer radius squared minus pi times inner radius squared. Integrate that difference over the interval. This handles shapes such as a pipe or a ring. If the outer radius is 3 and the inner radius is 1, each slice has area 8 pi.

## Related concepts

- [Work as an integral](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-9)
- [Area between curves](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-8)
- [Average value](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-9)
- [Integration by parts](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-1)

Web page: https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-6
Study this card in the app: https://studyvisually.com/app/learn?category=maths-university&card=maths-university-part-6-concept-6