Multiple Integrals

Prerequisites

Multiple integrals accumulate quantities over areas and volumes. They are the second module in Multivariable Calculus, after differential calculus and before vector calculus.

1. Meaning and bounds

For a surface density $\rho(x,y)$, mass over a region $R$ is

m=\iint_R\rho(x,y)\,dA.

For a volume density, mass over $E$ is

m=\iiint_E\rho(x,y,z)\,dV.

Over a rectangle, Fubini's theorem gives an iterated integral:

\iint_Rf\,dA=\int_a^b\int_c^d f(x,y)\,dy\,dx.

For a nonrectangular region, sketch first. Describe the same region with the reversed order before changing the bounds.

2. Average value and center of mass

If $A=\iint_R1\,dA$, the average value is

f_{\mathrm{avg}}=\frac1A\iint_Rf\,dA.

For a lamina,

\bar x=\frac1m\iint_Rx\rho\,dA, \qquad \bar y=\frac1m\iint_Ry\rho\,dA.

The same pattern produces moments and rotational inertias when the appropriate distance-squared factor is included.

3. Cylindrical and polar coordinates

For circular geometry,

x=r\cos\theta,\qquad y=r\sin\theta,\qquad dA=r\,dr\,d\theta.

The factor $r$ is the Jacobian determinant. For a disk of radius $a$,

\iint_{x^2+y^2\le a^2}f\,dA =\int_0^{2\pi}\int_0^a f(r\cos\theta,r\sin\theta)r\,dr\,d\theta.

In three dimensions, cylindrical coordinates add $z$ and use $dV=r\,dr\,d\theta\,dz$.

4. Spherical coordinates

Using $\phi$ measured from the positive $z$-axis,

x=\rho\sin\phi\cos\theta,\quad y=\rho\sin\phi\sin\theta,\quad z=\rho\cos\phi,

and

dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta.

For a full sphere, $0\le\rho\le a$, $0\le\phi\le\pi$, and $0\le\theta\le2\pi$. State the angle convention because some texts interchange $\theta$ and $\phi$.

5. General change of variables

If $(x,y)=(x(u,v),y(u,v))$, then

dA=\left|\frac{\partial(x,y)}{\partial(u,v)}\right|du\,dv, \qquad \frac{\partial(x,y)}{\partial(u,v)} =\det\begin{bmatrix}x_u&x_v\\y_u&y_v\end{bmatrix}.

In three dimensions use $|\det J|$ for volume. Check one-to-one behavior and multiplicity; the absolute value accounts for positive area or volume.

Checkpoint

You are ready for Vector Calculus when you can set up iterated bounds, choose coordinates from geometry, include the correct Jacobian, and interpret an integral as a physical total.