Multiple Integrals

Multiple integrals ( $\iint$ ) extend the concept of integration to functions of several variables. In algebra, they are tools for calculating volumes and analyzing continuous probability distributions.

Double Integrals

The double integral of

f (x, y)

over a region

R

is denoted

\iint_{R} f (x, y) d A

It represents the volume under the surface $z = f (x, y)$ above the region $R$ .

Fubini's Theorem

Theorem (Fubini's Theorem)

f

is continuous on a rectangle

R = [a, b] \times [c, d]

, then:

\iint_{R} f (x, y) d A = \int_{a}^{b} (\int_{c}^{d} f (x, y) d y) d x = \int_{c}^{d} (\int_{a}^{b} f (x, y) d x) d y

This allows us to evaluate double integrals as iterated integrals.

Change of Variables (Jacobian)

When changing variables (e.g., to polar coordinates), we must scale the area element

d A

by the determinant of the Jacobian matrix.

Polar Coordinates

x = r cos θ, y = r sin θ

d A = r d r d θ

\iint_{R} f (x, y) d x d y = \iint_{R^{'}} f (r cos θ, r sin θ) r d r d θ

Example (Gaussian Integral)

Prove

\int_{- \infty}^{\infty} e^{- x^{2}} d x = π

Proof. Let $I = \int_{- \infty}^{\infty} e^{- x^{2}} d x$ . $I^{2} = \int_{- \infty}^{\infty} e^{- x^{2}} d x \int_{- \infty}^{\infty} e^{- y^{2}} d y = \iint_{R^{2}} e^{- (x^{2} + y^{2})} d x d y$ . Switch to polar: $I^{2} = \int_{0}^{2 π} \int_{0}^{\infty} e^{- r^{2}} r d r d θ = 2 π [- \frac{1}{2} e^{- r^{2}}]_{0}^{\infty} = 2 π (1/2) = π$ . Thus $I = π$ . ∎

Practice Problems

Exercise (Problem 1)

Evaluate

\int_{0}^{1} \int_{0}^{x} e^{x^{2}} d y d x

. (Hint:

d y

integral is easy).

Exercise (Problem 2)

Find the volume of the region bounded by

z = 1 - x^{2} - y^{2}

and the

x y

-plane using polar coordinates.

Exercise (Problem 3)

Evaluate

\int_{0}^{1} \int_{y}^{1} sin (x^{2}) d x d y

by reversing the order of integration.