Theory

Double and triple integrals

How to integrate a function of two or three variables, how a multiple integral reduces to single integrals, how changes of variables simplify the domain, and how to compute areas, volumes, masses and centroids.

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The double integral over rectangles
Normal domains and reduction formulas

The double integral over rectangles

1. Double integral over a rectangle

Let $R = [a, b] \times [c, d]$ be a rectangle and let $f : R \to R$ be bounded. We subdivide $R$ into subrectangles $R_{ij}$ with a grid of points, choose a point $(ξ_{ij}, η_{ij})$ in each $R_{ij}$ and form the Riemann sum

S = i, j \sum f (ξ_{ij}, η_{ij}) Area (R_{ij}) .

If, as the dimensions of the subrectangles tend to $0$ , the sums tend to one and the same number for every choice of the points, $f$ is integrable on $R$ and that number is the double integral

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

Every function continuous on $R$ is integrable, and integrability is preserved if $f$ is continuous except on a finite number of regular curves. For $f \geq 0$ the integral is the volume of the solid lying between the rectangle $R$ and the graph of $f$ , just as in one variable the integral of a positive function is the area under the graph.

A subdivision of the rectangle

R

into subrectangles: in the colored subrectangle

R_{ij}

a point

(ξ_{ij}, η_{ij})

is chosen

2. Reduction theorem (Fubini) on rectangles

If $f$ is continuous on $R = [a, b] \times [c, d]$ , then

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

The theorem reduces the double integral to two successive single integrals: in the inner integral one variable is held fixed and we integrate with respect to the other, then the result, which is a function of a single variable, is integrated in turn. The proof is obtained by grouping the Riemann sums by columns or by rows, and we refer to standard textbooks for it. The two orders of integration give the same value, so we choose the order in which the computations are simpler.

Reduction on a rectangle

\iint_{[0, 1] \times [0, 2]} x y^{2} d x d y = \int_{0}^{1} (\int_{0}^{2} x y^{2} d y) d x = \int_{0}^{1} x \cdot \frac{8}{3} d x = \frac{8}{3} \cdot \frac{1}{2} = \frac{4}{3}

In the inner integral $x$ is a constant, therefore it comes out of the integral sign; what remains is $\int_{0}^{2} y^{2} d y = \frac{8}{3}$ .

3. Functions with separated variables

If $f (x, y) = g (x) h (y)$ with $g$ continuous on $[a, b]$ and $h$ continuous on $[c, d]$ , then

\iint_{[a, b] \times [c, d]} g (x) h (y) d x d y = (\int_{a}^{b} g (x) d x) (\int_{c}^{d} h (y) d y) .

It follows from Fubini’s theorem, since in the inner integral the factor that does not depend on the variable of integration comes out of the integral sign. The result holds only on rectangles: on a general domain the limits of the inner integral depend on the other variable and the factorization is not possible.

Normal domains and reduction formulas

1. Normal domain

A set $D \subseteq R^{2}$ is a normal domain with respect to the $x$ -axis if there exist two continuous functions $g_{1}, g_{2} : [a, b] \to R$ with $g_{1} \leq g_{2}$ such that

D = {(x, y) : {a \leq x \leq b g_{1} (x) \leq y \leq g_{2} (x)} .

The domain is normal with respect to the $y$ -axis if it is described in the same way with the roles exchanged: $c \leq y \leq d$ and $h_{1} (y) \leq x \leq h_{2} (y)$ . A domain normal with respect to $x$ is crossed by vertical segments that enter through the graph of $g_{1}$ and exit through the graph of $g_{2}$ ; this description directly provides the limits of integration. The disk and the triangle are normal with respect to both axes; many regions that are not normal can be split into a union of normal domains.

A domain normal with respect to the

x

-axis: the vertical segment at abscissa

x

enters through the graph of

g_{1}

and exits through the graph of

g_{2}

2. Reduction formulas on normal domains

Let $f$ be continuous on $D$ .

If $D$ is normal with respect to the $x$ -axis, with $a \leq x \leq b$ and $g_{1} (x) \leq y \leq g_{2} (x)$ , then

\iint_{D} f (x, y) d x d y = \int_{a}^{b} (\int_{g_{1} (x)}^{g_{2} (x)} f (x, y) d y) d x .

If $D$ is normal with respect to the $y$ -axis, with $c \leq y \leq d$ and $h_{1} (y) \leq x \leq h_{2} (y)$ , then

\iint_{D} f (x, y) d x d y = \int_{c}^{d} (\int_{h_{1} (y)}^{h_{2} (y)} f (x, y) d x) d y .

The outer variable ranges over a fixed interval, while the limits of the inner integral are functions of the outer variable. The first step of every exercise is therefore the description of the domain in normal form, with a sketch of the region to read off the functions $g_{1}$ and $g_{2}$ .

Domain between a parabola and a line

Let $D$ be the region lying between the parabola $y = x^{2}$ and the line $y = x$ . The two curves intersect where $x^{2} = x$ , that is, at $x = 0$ and $x = 1$ , and for $0 \leq x \leq 1$ we have $x^{2} \leq x$ . The domain is normal with respect to the $x$ -axis:

D = {(x, y) : {0 \leq x \leq 1 x^{2} \leq y \leq x}, \iint_{D} f d x d y = \int_{0}^{1} (\int_{x^{2}}^{x} f (x, y) d y) d x .

y = x^{2}

y = x

The domain lying between the parabola

y = x^{2}

and the line

y = x

: every vertical segment enters through the parabola and exits through the line

Changing the order of integration

When a domain is normal with respect to both axes, the two reduction formulas give the same value, therefore the order of integration can be changed by rewriting the domain in the other form. The change is useful when the given inner integral cannot be computed in elementary form: the function $e^{y^{2}}$ has no elementary antiderivative with respect to $y$ , so the integral $\int_{0}^{1} \int_{x}^{1} e^{y^{2}} d y d x$ is computed by integrating first with respect to $x$ . To change the order, we sketch the region described by the given limits and describe it again with respect to the other axis; the limits are not exchanged mechanically.

y = x

The region

0 \leq x \leq 1

x \leq y \leq 1

read in the two ways: the vertical segment goes from

y = x

y = 1

, the horizontal one goes from

x = 0

x = y

The theory continues with 6 more sections (properties of the double integral, change of variables, symmetries…), plus all worked examples, the solutions and the PDF.

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