22. Double Integrals over General Regions

We've seen how to compute $\large \iint_R f(x,y), dA$ if $\large R$ is a rectangular region. But how do we proceed if the region is something more general?

We will call regions like the following y-first regions (textbooks often call them y-simple regions). They go from some $\large y = g_1(x)$ below up to some $\large y=g_2(x)$ above and from some constant $\large x=a$ on the left to some $\large x=b$ on the right.

yfirst

^{^{Image from: https://courses.lumenlearning.com/calculus3/chapter/double-integrals-over-nonrectangular-regions}}

For a y-first region, we think of fixing $\large x$ at a constant value and integrating in $\large y$ to find the volume above a small sliver of area with width $\large dx$:

iterated1

^{^{Image from: https://instruct.math.lsa.umich.edu/lecturedemos/ma215/docs/15_3/examples.html}}

We then integrate the result in $\large x$ from $\large a$ to $\large b$ to add up all these little volumes:

iteratedGif

^{^{Image from: https://instruct.math.lsa.umich.edu/lecturedemos/ma215/docs/15_3/examples.html}}

The result is an iterated integral with variable endpoints on the inner integral:

$$\large \iint_R f(x,y)\, dA = \int_a^b\int_{g_1(x)}^{g_2(x)}f(x,y)\, dy dx$$

Here's a video showing how this is done:

Similarly, some regions are x-first or x-simple:

xfirst

^{^{Image from: https://courses.lumenlearning.com/calculus3/chapter/double-integrals-over-nonrectangular-regions/}}

We do the same thing, but our little slabs of volume are each at a constant $\large y$, and then we sweep through all the $\large y$ values:

iteratedGif2

^{^{Image from: https://courses.lumenlearning.com/calculus3/chapter/double-integrals-over-nonrectangular-regions/}}

In this case, we have:

$$\large \iint_R f(x,y)\, dA = \int_c^d\int_{h_1(y)}^{h_2(y)}f(x,y)\, dx dy$$

Some integrals can be done y-first or x-first, but on some problems, there is a clear preference to integrate first in a particular variable. In this video, we see that it is easier to set up the integral in a particular order:

In the example, the integral could be done either way, but it was easier to do it x-first. In some problems, an integral can only be worked out in one order. Because of this, it can be useful to change a given integral from y-first to x-first, or vice versa. Here's an example of how that is done:

The x/y-first order for a double integral could depend on the region of integration, on the integrand (the function we're integrating), or both! If something isn't working out in a particular order, it makes sense to try the other order.

For problems, we will need to be able to set up integrals, choose the best (or only) order, or change the order. It's fairly tricky and requires a bit of practice. The region of integration can be given as a plot, a description, or just as endpoints on integrals.

Remember, facts about double integrals are summarized on the Integral Fact Page.