How Does Leibniz' Integral Rule Optimize Constants in an Integral?

[adm_strat]

[SOLVED] Leibniz' Integral rule

Homework Statement

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.

Homework Equations

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

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

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.

 

[EnumaElish]

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

 

[adm_strat]

Duh, no I didn't integrate. Lol, I feel kinda dumb after that one. Thanks