Double Integral Applications — Theory
1. What double integrals measure
A double integral adds up small contributions over a two-dimensional region \(R\).
The basic form is:
\[
\iint_R f(x,y)\,dA.
\]
The meaning depends on the integrand \(f(x,y)\). If the integrand is \(1\), the integral gives area.
If the integrand is a height function, the integral gives volume. If the integrand is a density,
the integral gives mass.
2. Area of a region
The area of a plane region \(R\) is found by integrating \(1\) over the region:
\[
A=\iint_R 1\,dA.
\]
For a rectangular-type region
\[
a\le x\le b,
\qquad
g_1(x)\le y\le g_2(x),
\]
the area becomes:
\[
A=
\int_a^b
\int_{g_1(x)}^{g_2(x)}
1\,dy\,dx.
\]
3. Volume under a surface
Suppose a surface is given by
\[
z=h(x,y).
\]
The volume under this surface over the region \(R\) is:
\[
V=\iint_R h(x,y)\,dA.
\]
If \(h(x,y)\ge 0\), this represents the usual geometric volume between the surface
and the \(xy\)-plane.
4. Mass with variable density
A lamina is a thin flat plate. If its density varies from point to point, write the density as
\[
\rho=\rho(x,y).
\]
The total mass is:
\[
M=\iint_R \rho(x,y)\,dA.
\]
If the density is larger in one part of the region, that part contributes more to the total mass.
5. Moments of a lamina
The first moments measure how the mass is distributed relative to the coordinate axes.
The moment about the \(x\)-axis is:
\[
M_x=\iint_R y\,\rho(x,y)\,dA.
\]
The moment about the \(y\)-axis is:
\[
M_y=\iint_R x\,\rho(x,y)\,dA.
\]
The factor \(y\) appears in \(M_x\) because distance from the \(x\)-axis is measured vertically.
The factor \(x\) appears in \(M_y\) because distance from the \(y\)-axis is measured horizontally.
6. Center of mass
The center of mass is the balance point of the lamina. It is given by:
\[
\bar{x}=\frac{M_y}{M},
\qquad
\bar{y}=\frac{M_x}{M}.
\]
Notice the cross pattern:
- \(\bar{x}\) uses \(M_y\).
- \(\bar{y}\) uses \(M_x\).
This happens because \(M_y\) measures horizontal distribution and \(M_x\) measures vertical distribution.
7. Second moments
Second moments describe how far the mass is spread from the coordinate axes.
They are often used in physics and engineering.
\[
I_x=\iint_R y^2\rho(x,y)\,dA,
\qquad
I_y=\iint_R x^2\rho(x,y)\,dA.
\]
The polar second moment is:
\[
I_0=I_x+I_y
=
\iint_R (x^2+y^2)\rho(x,y)\,dA.
\]
8. Polar coordinates
Polar coordinates are useful for circular, sector, annular, and radial regions.
The conversion formulas are:
\[
x=r\cos\theta,
\qquad
y=r\sin\theta.
\]
The area element changes from \(dA\) to:
\[
dA=r\,dr\,d\theta.
\]
Therefore, a polar double integral has the form:
\[
\iint_R f(x,y)\,dA
=
\int_\alpha^\beta
\int_{r_1(\theta)}^{r_2(\theta)}
f(r\cos\theta,r\sin\theta)\,r\,dr\,d\theta.
\]
The factor \(r\) is essential. Forgetting it is one of the most common mistakes.
9. Example: mass of a lamina with \(\rho(x,y)=xy\)
Suppose \(R\) is the unit square:
\[
0\le x\le 1,
\qquad
0\le y\le 1.
\]
The mass is:
\[
M=
\int_0^1
\int_0^1
xy\,dy\,dx.
\]
First integrate with respect to \(y\):
\[
\int_0^1 xy\,dy
=
x\left[\frac{y^2}{2}\right]_0^1
=
\frac{x}{2}.
\]
Then integrate with respect to \(x\):
\[
M=
\int_0^1 \frac{x}{2}\,dx
=
\left[\frac{x^2}{4}\right]_0^1
=
\frac14.
\]
10. Example: center of mass for \(\rho(x,y)=xy\) on the unit square
The moment about the \(x\)-axis is:
\[
M_x=
\int_0^1
\int_0^1
y(xy)\,dy\,dx
=
\int_0^1
\int_0^1
xy^2\,dy\,dx
=
\frac16.
\]
The moment about the \(y\)-axis is:
\[
M_y=
\int_0^1
\int_0^1
x(xy)\,dy\,dx
=
\int_0^1
\int_0^1
x^2y\,dy\,dx
=
\frac16.
\]
Since \(M=\frac14\), the center of mass is:
\[
\bar{x}
=
\frac{M_y}{M}
=
\frac{1/6}{1/4}
=
\frac23,
\qquad
\bar{y}
=
\frac{M_x}{M}
=
\frac{1/6}{1/4}
=
\frac23.
\]
12. Common mistakes
- Forgetting the polar factor \(r\): in polar coordinates, \(dA=r\,dr\,d\theta\).
- Mixing up \(M_x\) and \(M_y\): \(M_x\) uses \(y\rho\), while \(M_y\) uses \(x\rho\).
- Using density for volume: volume uses height \(h(x,y)\), while mass uses density \(\rho(x,y)\).
- Using the wrong bounds: always sketch or inspect the region before integrating.
- Ignoring negative density: physical density should usually be nonnegative.
- Assuming midpoint results are exact: numerical integration gives approximations unless the case is especially simple.
- Choosing too few panels: curved regions and rapidly changing functions need more panels.
