| | 2025.01.28

5: Integral Calculus of Several Variables

Double and triple integrals for computing volumes, masses, and other quantities.

Double Integrals

Definition

\iint_R f(x, y) \, dA = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*, y_i^*) \Delta A_i

Iterated Integrals

Over a rectangle $R = [a, b] \times [c, d]$ :

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

Fubini’s Theorem: Order doesn’t matter if $f$ is continuous.

General Regions

Type I (bounded by $y = g_1(x)$ and $y = g_2(x)$ ):

\iint_D f(x, y) \, dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x, y) \, dy \, dx

Type II (bounded by $x = h_1(y)$ and $x = h_2(y)$ ):

\iint_D f(x, y) \, dA = \int_c^d \int_{h_1(y)}^{h_2(y)} f(x, y) \, dx \, dy

Applications of Double Integrals

Area

A = \iint_D 1 \, dA

Volume Under Surface

V = \iint_D f(x, y) \, dA

Mass and Center of Mass

For density $\rho(x, y)$ :

Mass: $m = \iint_D \rho(x, y) \, dA$

Center of mass:

\bar{x} = \frac{1}{m}\iint_D x\rho(x, y) \, dA, \quad \bar{y} = \frac{1}{m}\iint_D y\rho(x, y) \, dA

Moments of Inertia

I_x = \iint_D y^2 \rho \, dA, \quad I_y = \iint_D x^2 \rho \, dA, \quad I_0 = \iint_D (x^2 + y^2) \rho \, dA

Double Integrals in Polar Coordinates

When the region is circular or the integrand involves $x^2 + y^2$ :

\iint_R f(x, y) \, dA = \iint_R f(r\cos\theta, r\sin\theta) \, r \, dr \, d\theta

Key: $dA = r \, dr \, d\theta$ (not just $dr \, d\theta$ !)

Example: Disk of radius $a$

\iint_D f \, dA = \int_0^{2\pi} \int_0^a f(r\cos\theta, r\sin\theta) \, r \, dr \, d\theta

Triple Integrals

\iiint_E f(x, y, z) \, dV

Iterated Form

\iiint_E f \, dV = \int_a^b \int_{g_1(x)}^{g_2(x)} \int_{h_1(x,y)}^{h_2(x,y)} f(x, y, z) \, dz \, dy \, dx

Applications

Volume: $V = \iiint_E 1 \, dV$

Mass: $m = \iiint_E \rho(x, y, z) \, dV$

Center of mass: $\bar{x} = \frac{1}{m}\iiint_E x\rho \, dV$ , etc.

Cylindrical Coordinates

x = r\cos\theta, \quad y = r\sin\theta, \quad z = z

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

Use when: Region has circular symmetry about the $z$ -axis.

Example: Cylinder

\iiint_E f \, dV = \int_0^{2\pi} \int_0^a \int_0^h f(r\cos\theta, r\sin\theta, z) \, r \, dz \, dr \, d\theta

Spherical Coordinates

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

where:

$\rho$ = distance from origin
$\phi$ = angle from positive $z$ -axis ( $0 \leq \phi \leq \pi$ )
$\theta$ = angle in $xy$ -plane from positive $x$ -axis

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

Use when: Region has spherical symmetry.

Example: Sphere of radius $a$

\iiint_E f \, dV = \int_0^{2\pi} \int_0^{\pi} \int_0^a f \, \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta

Change of Variables (Jacobian)

For transformation $x = g(u, v)$ , $y = h(u, v)$ :

\iint_R f(x, y) \, dx \, dy = \iint_S f(g(u,v), h(u,v)) \left| \frac{\partial(x, y)}{\partial(u, v)} \right| du \, dv

Jacobian:

\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}

Common Jacobians

Coordinates	Jacobian
Polar	$r$
Cylindrical	$r$
Spherical	$\rho^2 \sin\phi$

Summary

Coordinate System	$dA$ or $dV$
Cartesian 2D	$dx \, dy$
Polar	$r \, dr \, d\theta$
Cartesian 3D	$dx \, dy \, dz$
Cylindrical	$r \, dz \, dr \, d\theta$
Spherical	$\rho^2 \sin\phi \, d\rho \, d\phi \, d\theta$