Multiple Integrals

Double Integrals

The double integral over a region $R$ in the $xy$-plane:

$\iint_R f(x, y)\,dx\,dy$

Iterated integrals: For rectangular region $a \leq x \leq b$, $c \leq y \leq d$:

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

Order can often be swapped (Fubini’s theorem).

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Triple Integrals

$\iiint_V f(x, y, z)\,dx\,dy\,dz$

The triple integral gives a volume when $f = 1$.

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Change of Variables

Under transformation $(x, y) \to (u, v)$:

$\iint_R f(x, y)\,dx\,dy = \iint_{R'} f(x(u,v), y(u,v))\,|J|\,du\,dv$

where the Jacobian is:

$J = \frac{\partial(x, y)}{\partial(u, v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}$

The absolute value $|J|$ accounts for the change in area element.

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Polar Coordinates

Transformation: $x = r\cos\theta$, $y = r\sin\theta$

Jacobian:

$J = \begin{vmatrix} \cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta \end{vmatrix} = r\cos^2\theta + r\sin^2\theta = r$

Area element: $dA = r\,dr\,d\theta$

Double integral:

$\iint_R f\,dA = \int_{\theta_1}^{\theta_2} \int_{r_1(\theta)}^{r_2(\theta)} f(r\cos\theta, r\sin\theta)\,r\,dr\,d\theta$

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Cylindrical Coordinates

$(x, y, z) \to (r, \theta, z)$ with $x = r\cos\theta$, $y = r\sin\theta$, $z = z$.

Volume element: $dV = r\,dr\,d\theta\,dz$

Useful for: cylinders, circular symmetry about $z$-axis.

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Spherical Coordinates

$(x, y, z) \to (r, \theta, \phi)$ with:

• $x = r\sin\theta\cos\phi$
• $y = r\sin\theta\sin\phi$
• $z = r\cos\theta$

where $r \geq 0$, $\theta \in [0, \pi]$, $\phi \in [0, 2\pi)$.

Jacobian: $J = r^2\sin\theta$

Volume element: $dV = r^2\sin\theta\,dr\,d\theta\,d\phi$

Derivation:

$J = \begin{vmatrix} \sin\theta\cos\phi & r\cos\theta\cos\phi & -r\sin\theta\sin\phi \\ \sin\theta\sin\phi & r\cos\theta\sin\phi & r\sin\theta\cos\phi \\ \cos\theta & -r\sin\theta & 0 \end{vmatrix} = r^2\sin\theta$

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Example: Volume of a Sphere

Using spherical coordinates:

$V = \iiint dV = \int_0^{2\pi} \int_0^\pi \int_0^R r^2\sin\theta\,dr\,d\theta\,d\phi$

$= \left[\frac{r^3}{3}\right]_0^R \cdot \left[-\cos\theta\right]_0^\pi \cdot \left[\phi\right]_0^{2\pi}$

$= \frac{R^3}{3} \cdot 2 \cdot 2\pi = \frac{4\pi R^3}{3}$

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Example: Gaussian Integral

Evaluate $I = \int_{-\infty}^{\infty} e^{-x^2}\,dx$.

Method: Square and convert to polar.

$I^2 = \int_{-\infty}^{\infty} e^{-x^2}\,dx \int_{-\infty}^{\infty} e^{-y^2}\,dy = \iint_{-\infty}^{\infty} e^{-(x^2+y^2)}\,dx\,dy$

Convert to polar: $x^2 + y^2 = r^2$, $dx\,dy = r\,dr\,d\theta$

$I^2 = \int_0^{2\pi} \int_0^{\infty} e^{-r^2} r\,dr\,d\theta = 2\pi \left[-\frac{1}{2}e^{-r^2}\right]_0^\infty = \pi$

Result:

$\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}$

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Choosing Coordinates

Geometry	Best Coordinates
Rectangular box	Cartesian
Cylinder, circular base	Cylindrical
Sphere, central force	Spherical
Ellipse	Modified polar or elliptical

Match the coordinate system to the domain boundaries for simplest limits.