# Leibniz' Integral rule

1. Dec 1, 2007

[SOLVED] Leibniz' Integral rule

1. The problem statement, all variables and given/known data

Use the Leibniz' integral rule for differentiating under the integral sign to determine constants a and b such that the integral $$\int^{1}_{0}(ax+b-x^{2})^{2} dx$$ is as small as possible.

2. Relevant equations

Leibniz' Interation was found at http://mathworld.wolfram.com/LeibnizIntegralRule.html" [Broken] and is as follows:

$$\frac{\partial}{\partial z} \int^{b(z)}_{a(z)} f(x,z) dx = \int^{b(z)}_{a(z)} \frac{\partial f}{\partial z} dx + f(b(z),z) \frac{\partial b}{\partial z} - f(a(z),z) \frac{\partial a}{\partial z}$$

3. The attempt at a solution

On my integral the limits of integration are constants and therefore the integral breaks down to:

$$\frac{\partial}{\partial z} \int^{b(z)}_{a(z)} f(x,z) dx = \int^{b(z)}_{a(z)} \frac{\partial f}{\partial z} dx$$

I took the partial with respect to x and I got:

$$f_{x} = 2 (ax +b - x^{2})(2x-a)$$=0

So two Equations result from that: $$ax+b-x^{2}=0$$ AND $$2x-a=0$$

Which gives: $$x=\frac{a}{2}$$

I substituted that into the other equation and got

$$b=\frac{a^{2}}{2}$$

I know this seems like some algebra somputations but I am really asking to see if I did the Leibniz integration underneath the integral correctly since I am stuck. Thanks for the help in advance.

Last edited by a moderator: May 3, 2017
2. Dec 1, 2007

### EnumaElish

Did you remember to integrate fx from 0 to 1 before setting the whole thing to zero?

3. Dec 1, 2007