Double And Triple Integral Evaluator — Theory
1. Double integrals
A double integral adds up values of a function over a two-dimensional region.
If the region is \(R\), then a double integral has the form
\[
\iint_R f(x,y)\,dA.
\]
In rectangular coordinates, \(dA=dx\,dy\) or \(dy\,dx\), depending on the chosen order.
2. Triple integrals
A triple integral adds up values of a function over a three-dimensional region.
If the solid region is \(E\), then a triple integral has the form
\[
\iiint_E f(x,y,z)\,dV.
\]
Triple integrals can represent volume, mass, total charge, total energy, and many other accumulated quantities.
3. Iterated integrals
A multiple integral is usually evaluated as an iterated integral.
For example,
\[
\int_a^b
\int_c^d
f(x,y)\,dy\,dx.
\]
The inner integral is evaluated first. In this example, the \(y\)-integral is evaluated first,
and the \(x\)-integral is evaluated second.
4. Rectangular coordinates
In rectangular coordinates, the area and volume elements are
\[
dA=dx\,dy,
\qquad
dV=dx\,dy\,dz.
\]
The Jacobian factor is \(1\), so no extra multiplier is needed.
5. Polar coordinates
Polar coordinates are useful for circular regions. They use
\[
x=r\cos\theta,
\qquad
y=r\sin\theta.
\]
The area element changes to
\[
dA=r\,dr\,d\theta.
\]
The extra factor \(r\) is the Jacobian.
6. Cylindrical coordinates
Cylindrical coordinates extend polar coordinates into three dimensions:
\[
x=r\cos\theta,
\qquad
y=r\sin\theta,
\qquad
z=z.
\]
The volume element is
\[
dV=r\,dr\,d\theta\,dz.
\]
Cylindrical coordinates are useful for cylinders, circular pipes, disks, and rotational solids.
7. Spherical coordinates
Spherical coordinates are useful for balls, shells, and radially symmetric regions.
They use
\[
x=\rho\sin\phi\cos\theta,
\qquad
y=\rho\sin\phi\sin\theta,
\qquad
z=\rho\cos\phi.
\]
The volume element is
\[
dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta.
\]
The factor \(\rho^2\sin\phi\) is the spherical-coordinate Jacobian.
8. Order switching
The order of integration can often be changed, but the bounds may also need to change.
For a rectangular box with constant bounds, changing the order usually does not change the bounds.
For a non-rectangular region, changing the order usually requires rewriting the region.
\[
\int_a^b
\int_{g_1(x)}^{g_2(x)}
f(x,y)\,dy\,dx
\]
In this setup, the \(y\)-bounds depend on \(x\). If the order is switched, the new \(x\)-bounds may depend on \(y\).
9. Example: unit cube
Consider
\[
\iiint_E xyz\,dV,
\qquad
0\le x\le 1,\quad 0\le y\le 1,\quad 0\le z\le 1.
\]
The integral separates:
\[
\begin{aligned}
\int_0^1\int_0^1\int_0^1 xyz\,dz\,dy\,dx
&=
\left(\int_0^1 x\,dx\right)
\left(\int_0^1 y\,dy\right)
\left(\int_0^1 z\,dz\right)\\
&=
\frac12\cdot\frac12\cdot\frac12\\
&=
\frac18.
\end{aligned}
\]
11. Common mistakes
- Forgetting the Jacobian: polar and cylindrical integrals need \(r\); spherical integrals need \(\rho^2\sin\phi\).
- Using the wrong angle bounds: full circular rotation is usually \(0\le\theta\le 2\pi\).
- Switching order without changing bounds: non-rectangular regions often require new bounds after switching order.
- Confusing \(r\) and \(\rho\): \(r\) is cylindrical radius, while \(\rho\) is spherical distance from the origin.
- Using degrees instead of radians: trigonometric bounds should be in radians, such as \(\pi/2\) or \(2\pi\).
