Topic 13 / 17advanced

Double & Triple Integrals

Set up and evaluate double and triple integrals as iterated integrals; find area, volume, mass, and average value; switch the order of integration.

A double integral $\iint_{R} f d A$ accumulates values of $f$ over a region $R$ in the plane. A triple integral $∭_{E} f d V$ does the same in 3D. Computation reduces to **iterated integrals** — nested single integrals with carefully chosen bounds.

Iterated integrals over rectangles

Over

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

\iint_{R} f 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 .

Fubini's theorem says order doesn't matter for continuous

f

on a rectangle.

General regions

Type I (vertical slices):

a \leq x \leq b, g_{1} (x) \leq y \leq g_{2} (x)

— integrate

y

first. Type II (horizontal slices):

c \leq y \leq d, h_{1} (y) \leq x \leq h_{2} (y)

— integrate

x

first. The **inner** bounds may depend on the outer variable; the **outer** bounds must be constants.

Switching the order of integration

Sometimes one order is impossible (e.g., the integrand has no closed-form antiderivative in one variable). Sketch the region, then re-describe it in the other type. Re-write bounds; the integrand stays the same.

Strategy tips

**Always sketch the region** before changing order.
Look for triangular or wedge-shaped regions where one order has neat bounds.

Applications: area, volume, mass, average value

Area of

R

A = \iint_{R} d A

. Volume of solid

E

V = ∭_{E} d V

. Volume between graph

z = f (x, y)

and the

x y

-plane:

V = \iint_{R} f d A

(with

f \geq 0

). Mass with density

ρ

m = \iint_{R} ρ d A

(or triple). Average value:

\overset{ˉ}{f} = \frac{1}{A} \iint_{R} f d A

Triple integrals

∭_{E} f d V

. Six orderings:

d x d y d z, d y d x d z, \dots

Choose the easiest. For type I in

z

z

between two surfaces

z_{1} (x, y)

and

z_{2} (x, y)

, with

(x, y)

in projection

D

Worked examples

Example 1

Evaluate

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

1
Inner integral, $x$ from 0 to 2, $y$ constant.
$\int_{0}^{2} (y + 2 x e^{y}) d x = [y x + x^{2} e^{y}]_{0}^{2} = 2 y + 4 e^{y}$
2
Outer integral.
$\int_{1}^{2} (2 y + 4 e^{y}) d y = [y^{2} + 4 e^{y}]_{1}^{2} = (4 + 4 e^{2}) - (1 + 4 e) = 3 + 4 e^{2} - 4 e$

Answer

3 + 4 e^{2} - 4 e

Example 2

Switch order of integration and evaluate

\int_{0}^{1} \int_{y}^{1} \frac{y e ^{x^{2}}}{x ^{3}} d x d y

1
Sketch region. Bounds $y \leq x \leq 1$ and $0 \leq y \leq 1$ describe $0 \leq y \leq x^{2}, 0 \leq x \leq 1$ in the other order.
2
Switch.
$\int_{0}^{1} \int_{0}^{x^{2}} \frac{y e ^{x^{2}}}{x ^{3}} d y d x$
3
Inner integral.
$\int_{0}^{x^{2}} \frac{y e ^{x^{2}}}{x ^{3}} d y = \frac{e ^{x^{2}}}{x ^{3}} \cdot \frac{x ^{4}}{2} = \frac{x e ^{x^{2}}}{2}$
4
Outer.
$\int_{0}^{1} \frac{x e ^{x^{2}}}{2} d x = \frac{e ^{x^{2}}}{4}_{0}^{1} = \frac{e - 1}{4}$

Answer

(e - 1) /4

Example 3

Find the volume of the tetrahedron with vertices

(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)

1
Plane through nonorigin vertices: $x + y + z = 1$ .
2
Set up triple integral.
$V = \int_{0}^{1} \int_{0}^{1 - x} \int_{0}^{1 - x - y} d z d y d x$
3
Innermost.
$= \int_{0}^{1} \int_{0}^{1 - x} (1 - x - y) d y d x$
4
Middle.
$= \int_{0}^{1} [(1 - x) y - \frac{y ^{2}}{2}]_{0}^{1 - x} d x = \int_{0}^{1} \frac{( 1 - x ) ^{2}}{2} d x$
5
Outer.
$= \frac{( 1 - x ) ^{3}}{- 6}_{0}^{1} = \frac{1}{6}$

Answer

V = 1/6

Interactive visualizations

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Tetrahedron $\{(x, y, z) : x, y, z \ge 0, x + y + z \le 1\}$ — the region for the worked example.

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Solid bounded above by $z = 4$, below by paraboloid $z = x^2 + y^2$.

Formulas in this topic

Area

A = \iint_{R} d A

Region area as a double integral.

Volume (triple)

V = ∭_{E} d V

Solid volume.

Mass with density

m = \iint_{R} ρ d A

Or triple for solids.

Average value

\overset{ˉ}{f} = \frac{1}{A} \iint_{R} f d A

Average over the region.

Iterated integrals over rectangles

General regions

Switching the order of integration

Applications: area, volume, mass, average value

Triple integrals

Worked examples

Interactive visualizations

Formulas in this topic

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