Recall our definition of the definite integral of a function of a single variable:

Let f(x) be defined on [a,b] and let x₀,x₁,¼,x_n be a partition of [a,b]. For each [x_i-1,x_i], let x_i^* Î [x_i-1,x_i]. Then

ó õ b a  f(x) dx = lim maxDxi ® 0  n å i = 1  f(xi*)Dxi

Take a quick look at the Riemann Sum Tutorial

We can extend this definition to define the integral of a function of two or more variables.

Double Integral of a Function of Two Variables

Let f(x,y) be defined on a closed and bounded region R of the xy-plane. Set up a grid of vertical and horizontal lines in the xy-plane to form an inner partition of R into n rectangular subregions R_k of area DA_k, each of which lies entirely in R. (Ignore the rectangles that are not entirely contained in R) Choose a point (x_k^*, y_k^*) in each subregion R_k. The sum

• If this limit exists, we say that f is integrable over the region of integration R.

• If f is continuous on R, then f is integrable over R.

Geometric Interpretation of the Double Integral

Notice that as we increase the density of our grid, the sum

n å k=1

Ak of the individual rectangles better and better

approximates the area of region R. In the limit as DA_k ® 0, we have

The interpretaion of the double integral as a volume still holds if f(x,y) takes on both positive and negative values. In this case, we obtain the difference between the volume above the xy-plane between z = f(x,y) and R and the volume below the xy-plane between z = f(x,y) and R.

We next turn to the actual evaluation of double integrals.

Iterated Integrals

In the double integral

òò R

f(x,y) dA, dA may be viewed informally as an infinitesimal

area of a rectangle inside R with dimensions dy and dx. For the kinds of ``ordinary''functions and regions we'll be concerned with,

where the limits of integration are determined by the region R over which we are integrating.

These integrals are called iterated integrals, since we integrate more than once.

	We integrate ``from the inside out.'' That is, in ò	b a	ò	g2(x) g1(x)	f(x,y) dy dx, we first integrate
	f(x,y) with respect to y and evaluate it at g2(x) and g1(x). We then integrate the result with respect to x and evaluate the outcome at a and b.

Iterated triple integrals

òòò G

f(x,y,z)dV can be defined in a similar way.

An example will make these ideas more concrete.

Example

Let's evaluate the double integral

òò R

6xy dA, where R is the region bounded by

y = 0, x = 2, and y = x². We will verify here that the order of integration is unimportant:

óó õõ R 6xy dA = ó õ 2 0  ó õ x2 0  6xy dy dx = ó õ 2 0  [ 3xy2 |y = 0x2 ] dx = ó õ 2 0  3x5 dx = 1 2 x6 ê ê ê 2 x = 0  = 1 2 (64)- 1 2 (0) = 32

óó õõ R 6xy dA = ó õ 4 0  ó õ 2 Öy  6xy dx dy = ó õ 4 0  [ 3x2y |x = Öy2 ] dy = ó õ 4 0  ( 12y-3y2 ) dy = ( 6y2 -y3 ) |4y = 0 = ( 6(4)2-(4)3 ) - ( 6(0)2- (0)3 ) = 32

so

òò R

6xy dA = 32 here, regardless of the order in which we carry out

the integration, as long as we are careful to set up the limits of integration correctly.

Now for a triple integral...

Example

We will evaluate the triple integral ò

2 0

ò

y2 -1

ò

z 1

yz dx dz dy.

Let f(x,y) be defined on a closed and bouned region R of the xy-plane. Then

We evaluate

òò R

f(x,y) dA as an iterated integral: