Multiple Integrals

Level 1 - Calculus topic page in Several-Variable Calculus.

Principle

Multiple integrals accumulate a function over a two-dimensional or three-dimensional region. They generalise area and volume integrals.

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

double integral over region R

\(R\)

region of integration

\(dA\)

small area element

For rectangular regions, integrate one variable at a time.

Iterated integral

\[\iint_R f(x,y)\,dA=\int_a^b\int_c^d f(x,y)\,dy\,dx\]

The inner integral treats the outer variable as constant. For non-rectangular regions, the limits describe the boundary curves.

Question

Evaluate \(\int_0^1\int_0^2 xy\,dy\,dx\).

Answer

First integrate with respect to \(y\): \(\int_0^2 xy\,dy=x[y^2/2]_0^2=2x\). Now integrate with respect to \(x\): \(\int_0^1 2x\,dx=[x^2]_0^1=1\).

Question

What does \(\iint_R 1\,dA\) represent?

Answer

Integrating the constant function \(1\) over a region adds one unit of height over every small area element. The result is the area of \(R\).