The Unapologetic Mathematician

Mathematics for the interested outsider

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 f:R\rightarrow\mathbb{R} is a continuous function on the rectangle [a,b]\times[c,d], and that D_2f is also continuous on R. Also, let p:[c,d]\rightarrow[a,b] and q:[c,d]\rightarrow[a,b] be two differentiable functions on [c,d] with images in [a,b]. Define the function

\displaystyle F(y)=\int\limits_{p(y)}^{q(y)}f(x,y)\,dx

Then the derivative F'(y) exists and has the value

\displaystyle F'(y)=\int\limits_{p(y)}^{q(y)}\left[D_2f\right](x,y)\,dx+f(q(y),y)q'(y)-f(p(y),y)p'(y)

In fact, if we forget letting f depend on y 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

\displaystyle G(x^1,x^2,x^3)=\int\limits_{x^1}^{x^2}f(t,x^3)\,dt for x^1 and x^2 in [a,b] and x^3 in [c,d]. Then F(y)=G(p(y),q(y),y).

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

\displaystyle\begin{aligned}\frac{\partial G}{\partial x^1}&=-f(x^1,x^3)\\\frac{\partial G}{\partial x^2}&=f(x^2,x^3)\\\frac{\partial G}{\partial x^3}&=\int\limits_{x^1}^{x^2}\left[D_2f\right](t,x^3)\,dt\end{aligned}

Then we can use the chain rule:

\displaystyle\begin{aligned}\frac{d}{dy}F(y)&=\frac{d}{dy}G(p(y),q(y),y)\\&=\frac{\partial G}{\partial x^1}\frac{dp}{dy}+\frac{\partial G}{\partial x^2}\frac{dq}{dy}+\frac{\partial G}{\partial x^3}\frac{dy}{dy}\\&=-f(p(y),y)p'(y)+f(q(y),y)q'(y)+\int\limits_{p(y)}^{q(y)}\left[D_2f\right](x,y)\,dx\end{aligned}

January 14, 2010 Posted by | Analysis, Calculus | 3 Comments