## Differentiating Partial Integrals

A neat little variation on differentiating an integral from last time combines it with the fundamental theorem of calculus. It’s especially interesting in the context of evaluating iterated integrals for irregular regions where the limits of integration may depend on other variables.

Let is a continuous function on the rectangle , and that is also continuous on . Also, let and be two differentiable functions on with images in . Define the function

Then the derivative exists and has the value

In fact, if we forget letting depend on at all, this is the source of some of my favorite questions on first-semester calculus finals.

Anyway, we define another function for the moment

for and in and in . Then .

The fundamental theorem of calculus tells us the first two partial derivatives of immediately, and for the third we can differentiate under the integral sign:

Then we can use the chain rule:

Please allow me to write here an easy exercise (in a different notation of yours).

.

Exercise. Find the derivative of the integral

Solution. In this case we have and The derivatives are

.

The values of the integrand function are evaluated at and

Hence

Comment by Américo Tavares | January 15, 2010 |

You keep writing out these “exercises” in the comments.. why not make posts about them and link here? I know you have your own weblog, and you aren’t really commenting at all.

Comment by John Armstrong | January 15, 2010 |

I wrote them here because I thought they were useful and appropriate. Since that is not the case, I will stop writing them as comments here. Of course you can delete them all.

Comment by Américo Tavares | January 15, 2010 |