Multiple Integrals

How accumulation extends from intervals to regions in the plane and space, why double and triple integrals are limits of sums over many-variable domains, and how densities, mass, and expectation fit into the picture.
Modified

April 26, 2026

Keywords

double integral, triple integral, iterated integral, density, volume, multivariable calculus

1 Role

This page is the accumulation page of multivariable calculus.

Its job is to extend the integral idea from one-dimensional intervals to two- and three-dimensional regions, so that mass, volume, probability, and other accumulated quantities can be expressed over domains in the plane or in space.

2 First-Pass Promise

Read this page after Integrals and Accumulation.

If you stop here, you should still understand:

  • why double and triple integrals are limits of sums over regions
  • how iterated integrals implement accumulation one variable at a time
  • why density is the natural interpretation for many multiple integrals
  • how multiple integration connects to probability and geometric modeling

3 Why It Matters

Single-variable integration accumulates across an interval.

But many real quantities live on regions, not just on lines:

  • mass spread over a plate
  • probability density over a plane
  • energy over a volume
  • average temperature over a region
  • accumulated objective contributions over many-variable states

Multiple integrals are the language for these problems.

This matters because later probability, scientific computing, and geometry all rely on the same idea:

small local density × small local region element, summed across the whole domain.

So this page is where the accumulation branch of multivariable calculus becomes as central as the derivative branch.

4 Prerequisite Recall

  • a definite integral is a limit of Riemann sums
  • multiple variables require describing a region, not just an interval
  • Jacobian and linearization ideas later explain why coordinate changes work, but the first-pass story is still accumulation by small pieces

5 Intuition

For a function \(f(x,y)\) over a region \(R\) in the plane, imagine cutting \(R\) into many tiny rectangles.

Over one tiny piece with area \(\Delta A\), the contribution is approximately

\[ f(x_i,y_i)\Delta A. \]

Add all the pieces and refine the partition:

\[ \sum_i f(x_i,y_i)\Delta A_i. \]

If this stabilizes as the pieces get finer, the limit is the double integral

\[ \iint_R f(x,y)\,dA. \]

The same idea works in space:

\[ \iiint_E f(x,y,z)\,dV. \]

So conceptually, nothing mystical changed. The integral is still a limit of local contributions. Only the geometry of the domain became richer.

6 Formal Core

Definition 1 (Double Integral) For a suitable function \(f(x,y)\) on a planar region \(R\),

\[ \iint_R f(x,y)\,dA \]

represents the accumulated contribution of \(f\) over the area region \(R\).

At a first-pass level, it is the limit of Riemann sums over finer and finer partitions whose largest piece size goes to zero.

Definition 2 (Triple Integral) For a suitable function \(f(x,y,z)\) on a spatial region \(E\),

\[ \iiint_E f(x,y,z)\,dV \]

represents the accumulated contribution of \(f\) over the volume region \(E\).

Again, the conceptual definition is a limit of sums of local density values times small volume elements over finer and finer partitions whose largest piece size goes to zero.

Definition 3 (Iterated Integral) Many multiple integrals are computed as repeated one-variable integrals, for example

\[ \iint_R f(x,y)\,dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y)\,dy\,dx. \]

This is an iterated integral: accumulate in one variable first, then in the other.

Proposition 1 (Density Interpretation) If \(f\) is a density:

  • over area, \(\iint_R f\,dA\) gives total mass or total probability over \(R\)
  • over volume, \(\iiint_E f\,dV\) gives total mass or total probability over \(E\)

This is one of the most important applied interpretations of multiple integrals.

7 Worked Example

Suppose a thin rectangular plate occupies

\[ R=[0,2]\times[0,1] \]

and has density

\[ \rho(x,y)=x+y. \]

Its total mass is

\[ M=\iint_R (x+y)\,dA. \]

Write this as an iterated integral:

\[ M=\int_0^2 \int_0^1 (x+y)\,dy\,dx. \]

Integrate with respect to \(y\) first:

\[ \int_0^1 (x+y)\,dy = \left[xy+\frac{y^2}{2}\right]_0^1 =x+\frac12. \]

Now integrate with respect to \(x\):

\[ M=\int_0^2 \left(x+\frac12\right)\,dx = \left[\frac{x^2}{2}+\frac{x}{2}\right]_0^2 =2+1=3. \]

So the total mass of the plate is

\[ M=3. \]

This example shows the standard template:

  • identify the region
  • identify the density
  • write the multiple integral
  • evaluate it as an iterated integral

8 Computation Lens

A practical first-pass workflow for multiple integrals is:

  1. decide what region is being integrated over
  2. decide what quantity is being accumulated: volume, mass, probability, average, or something else
  3. write the integral with the right geometric element (\(dA\) or \(dV\))
  4. express it as an iterated integral using bounds that describe the region
  5. compute one variable at a time

The main difficulty is often not the calculus itself, but describing the region correctly.

9 Application Lens

Multiple integrals matter immediately in several parts of the site:

  • in probability, densities over several variables must integrate to total probability \(1\)
  • in statistics, expected values and likelihood normalizations often hide multivariable integrals
  • in engineering, mass, charge, energy, and average-value models are often region integrals
  • in optimization and ML, change-of-variables ideas and continuous models often depend on this viewpoint

So this page gives the accumulation language needed for high-dimensional modeling, not just for geometric classroom problems.

10 Stop Here For First Pass

If you can now explain:

  • why a multiple integral is still a limit of local contributions
  • how a double integral differs from an iterated integral only in viewpoint, not in meaning
  • why density is a natural interpretation
  • why region description is central

then this page has done its main job.

11 Go Deeper

The strongest next steps after this page are:

  1. Vector Fields and Divergence / Curl, for the field-based branch of multivariable calculus
  2. Probability, to see multi-variable density and expectation ideas in a probabilistic setting
  3. Optimization, to see how integration appears in continuous objectives and modeling

12 Optional Deeper Reading

13 Optional After First Pass

If you want more practice before moving on:

  • rewrite a double integral in two different valid orders of integration
  • take a density function and interpret the integral as total mass or total probability
  • sketch a region before writing any bounds

14 Common Mistakes

  • writing an iterated integral before understanding the region
  • treating \(dA\) or \(dV\) as decorative instead of geometric information
  • forgetting whether the integrand is a height, a density, or some other local quantity
  • mixing up volume under a surface with total mass from a density
  • assuming order changes are trivial before checking the geometry

15 Sources and Further Reading

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