Multiple Integrals

By SkytrailGroup Mathematics Editorial Team April 12, 2026

Double Integrals

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

Examples

Example 1. \(\int_0^1\int_0^1 xy\,dy\,dx\)

Solution. = \(\int_0^1 x/2\,dx=1/4\).

In Depth

Multiple integrals extend the definite integral to functions of several variables. The double integral \(\iint_R f(x,y)\,dA\) computes the signed volume under the surface \(z=f(x,y)\) over the region \(R\). Fubini's theorem: for continuous \(f\) on a rectangle, \(\iint_R f\,dA = \int_a^b\int_c^d f(x,y)\,dy\,dx\).

For non-rectangular regions, the limits of the inner integral depend on the outer variable. Choosing the order of integration wisely can simplify the computation dramatically. Switching order requires re-describing the region.

Change of variables in double integrals: \(\iint_R f(x,y)\,dA = \iint_S f(x(u,v),y(u,v))|J|\,du\,dv\) where \(J = \partial(x,y)/\partial(u,v)\) is the Jacobian determinant. Polar coordinates (\(x=r\cos\theta\), \(y=r\sin\theta\), \(|J|=r\)) simplify integrals over circular regions.

Triple integrals \(\iiint_V f\,dV\) compute mass, charge, and other quantities distributed in 3D space. Cylindrical coordinates (\(|J|=r\)) suit problems with axial symmetry; spherical coordinates (\(|J|=\rho^2\sin\phi\)) suit problems with spherical symmetry.

Key Properties & Applications

Double integrals \(\iint_R f(x,y)\,dA\) compute volumes, masses, and averages over 2D regions. Fubini's theorem allows evaluation as iterated integrals: \(\iint_R f\,dA=\int_a^b\int_{g_1(x)}^{g_2(x)}f(x,y)\,dy\,dx\). The order of integration can be switched (with adjusted limits) to simplify computation.

Change of variables uses the Jacobian: \(\iint_R f(x,y)\,dA=\iint_S f(x(u,v),y(u,v))|J|\,du\,dv\) where \(J=\partial(x,y)/\partial(u,v)\). Polar coordinates (\(J=r\)), cylindrical (\(J=r\)), and spherical (\(J=\rho^2\sin\phi\)) are the most common transformations.

Triple integrals extend to 3D: \(\iiint_V f\,dV\) computes mass, charge, or probability over a 3D region. Applications include computing moments of inertia, centers of mass, and gravitational potentials. The divergence theorem converts volume integrals to surface integrals, often simplifying computation.