Integral Calculus of Several Variables

This chapter systematically studies multiple integrals, including double integrals, triple integrals, line integrals, surface integrals, and their applications in geometry and physics.

Double and Triple Integrals

Definition and Properties of Double Integrals

$\iint_D f(x, y) dA$
Computation in rectangular and polar coordinates
Linearity, additivity over regions

Definition and Properties of Triple Integrals

$\iiint_\Omega f(x, y, z) dV$
Computation in rectangular, cylindrical, and spherical coordinates

Applications of Double and Triple Integrals

Area, volume, mass, and physical quantities

Line Integrals and Surface Integrals

First Kind Line Integral

$\int_C f(x, y) ds$
Used to compute curve length, mass, etc.

Second Kind Line Integral

$\int_C P dx + Q dy$
Physical applications: work, flow

Surface Integrals

$\iint_S f(x, y, z) dS$
$\iint_S P dy dz + Q dz dx + R dx dy$

Important Formulas

Green’s Theorem

$\iint_D (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dA = \int_{\partial D} P dx + Q dy$

Gauss’s Theorem (Divergence Theorem)

$\iiint_\Omega \nabla \cdot \vec{F} dV = \iint_{\partial \Omega} \vec{F} \cdot d\vec{S}$

Stokes’ Theorem

$\iint_S (\nabla \times \vec{F}) \cdot d\vec{S} = \int_{\partial S} \vec{F} \cdot d\vec{r}$

Exercises

Compute the double integral $\iint_D x^2 y dA$ , where $D$ is $0 \leq x \leq 1, 0 \leq y \leq 2$ .
Compute the triple integral $\iiint_\Omega z dV$ , where $\Omega$ is $0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 2$ .
Compute the line integral $\int_C (x^2 + y^2) ds$ , where $C$ is the unit circle $x^2 + y^2 = 1$ .
Use Green’s theorem to compute $\int_C (x - y) dx + (x + y) dy$ , where $C$ is the positively oriented circle centered at the origin with radius 1.
Compute the surface integral $\iint_S z dS$ over the sphere $x^2 + y^2 + z^2 = 1$ .

Reference Answers

1. Compute the double integral $\iint_D x^2 y dA$

$\int_0^1 \int_0^2 x^2 y dy dx = \int_0^1 x^2 [\frac{1}{2}y^2]_0^2 dx = \int_0^1 x^2 \times 2 dx = 2 \int_0^1 x^2 dx = 2 \times \frac{1}{3} = \frac{2}{3}$

2. Compute the triple integral $\iiint_\Omega z dV$

$\int_0^1 \int_0^1 \int_0^2 z dz dy dx = \int_0^1 \int_0^1 [\frac{1}{2}z^2]_0^2 dy dx = \int_0^1 \int_0^1 2 dy dx = 2$

3. Compute the line integral $\int_C (x^2 + y^2) ds$

$x^2 + y^2 = 1$ , $ds$ along the unit circle, the value is $2\pi$

4. Use Green’s theorem to compute $\int_C (x - y) dx + (x + y) dy$

$\frac{\partial (x + y)}{\partial x} - \frac{\partial (x - y)}{\partial y} = 1 - (-1) = 2$

The area is $\pi$ , so the value is $2\pi$

5. Compute the surface integral $\iint_S z dS$ over the sphere $x^2 + y^2 + z^2 = 1$

The sphere is symmetric, $z$ is an odd function about $z=0$ , so the integral is $0$