Home page for accesible maths 4 Double integrals: general regions and change of variable 4 Double integrals: general regions and change of variable 4.2 Double integrals over triangular and more general regions

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4.1 Review of double integrals

Let

R

be a region in the

(x,y)

-plane,

f

a function of

x

and

y

. The double integral

I=\int\!\int_{R}f(x,y)\;dx\;dy

is the volume between the region

R

and the surface

z=f(x,y)

, with any volume below the

(x,y)

-plane counted negatively.

This is a definite integral, with the region $R$ taking the place of the interval $[a,b]$ occurring in the one-dimensional case.

Double integrals can be calculated by two repetitions of ordinary integration, as follows.

First consider the case where

R

is a rectangle, given by:

a\leq x\leq b

c\leq y\leq d

. For a fixed value of

y

, the cross-section of our volume is the “wall” shown in the picture; this is the area under the curve

z=f(x,y)

in this plane, so it equals

\int_{a}^{b}f(x,y)\;dx

(in which the integration is with respect to $x$ , with $y$ treated as constant). The contribution to our volume of the slice between $y$ and $y+\delta y$ is roughly $\delta y$ times this integral. Hence

I=\int_{c}^{d}\left(\int_{a}^{b}f(x,y)\;dx\right)dy.

(7)

By considering cross-sections for fixed $x$ instead, we see that also

I=\int_{a}^{b}\left(\int_{c}^{d}f(x,y)\;dy\right)dx.

(8)

(7) and (8) are called repeated integrals. They are commonly written without the brackets: the meaning is the same as if the brackets were there, so (8) becomes

\int_{a}^{b}\int_{c}^{d}f(x,y)\;dy\;dx.

Note. The double integral $\int\!\int_{R}1\>dx\;dy$ is simply the area of $R$ .