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.
- 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 .
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 .
 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
Stephanie Glen. "Multiple Integrals: Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/multiple-integrals/
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