 # Multiple Integrals: Definition, Examples

Share on

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

1. Double Integrals
2. Triple Integrals
3. 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 A solution for a triple integral.

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:

1. 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.
2. Sample one point in each box, multiplying f at the point * the volume of the box,
3. Sum up over all boxes.

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

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 .

## References

 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
 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!