**Multiple integrals** are definite integrals and are an extension of the ordinary (single) integral, which finds an area below a curve. Usually, the focus is on double integrals, which find the area of a region in a plane, and triple integrals which find volumes.

Multiple integrals arise in many areas of physical science, especially in mechanics to find volumes, masses, and moments of inertia.

## Types of Multiple Integrals

- Double Integrals
- Triple Integrals
- Multiple Integrals for Higher Dimensions

## 1. Double Integrals

Double integrals allow you to find the volume for a two-dimensional area or surface. General steps to solving:

- Evaluate the inner integral,
- Substitute the result into the equation,
- Solve the outer integral.

For a step by step example, see: Double Integral Worked Example.

## 2. Triple Integrals

**Triple integrals**extend integration to find 3D volumes or mass, when the volume has variable density. For a worked example, see: Center of mass with triple integrals.

## 3. Multiple Integrals for Higher Dimensions

It’s theoretically possible to have a multiple integral for any number of dimensions. For example, **quadruple integrals** find *hypervolumes*: the volume of objects in four-dimensional space.

Let’s say we had a 4-D box in a space with coordinates (x, y, z, w). the quadruple integral could be defined in the same way as integrals in lower dimensions are defined: as limits of Riemann sums:

- Subdivide the box into
*i j k l*equal-sized boxes:*i*boxes in the x direction,*j*boxes in the y direction,*k*boxes in the z direction,*l*boxes in the w direction.

- Sample one point in each box, multiplying f at the point * the volume of the box,
- Sum up over all boxes.

The quadruple integral is limit of *i j k l* as they go to infinity [1].

Integrals in higher dimensions are rarely seen as they have little practical use and are challenging to calculate. For example the stream depletion rate (in hydrology) can be expressed as a quintuple integral, but the solution requires less computational effort when computing four of the integrals analytically [2].

## References

[1] Rowan, J. (2018). Math 53 Summer 2018 Homework Assignment 20: Solutions to selected problems. Retrieved May 5, 2021 from: https://math.berkeley.edu/~jrowan/53Summer18/MATH53Su18ROWAN-ASSIGNMENT20SOLNS.pdf

[2] Maroney, C. & Rehmann, C. (2017). Stream depletion rate for a radial collector well in an unconfined aquifer near a fully penetrating river. Journal of Hydrology, Volume 547, April, Pages 732-741

**CITE THIS AS:**

**Stephanie Glen**. "Multiple Integrals: Definition, Examples" From

**CalculusHowTo.com**: Calculus for the rest of us! https://www.calculushowto.com/multiple-integrals/

**Need help with a homework or test question? **With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!