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

Use the Leibniz' integral rule for differentiating under the integral sign to determine constantsaandbsuch that the integral [tex]\int^{1}_{0}(ax+b-x^{2})^{2} dx [/tex] is as small as possible.

2. Relevant equations

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

[tex]\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} [/tex]

3. The attempt at a solution

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

[tex]\frac{\partial}{\partial z} \int^{b(z)}_{a(z)} f(x,z) dx = \int^{b(z)}_{a(z)} \frac{\partial f}{\partial z} dx [/tex]

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

[tex]f_{x} = 2 (ax +b - x^{2})(2x-a)[/tex]=0

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

Which gives: [tex]x=\frac{a}{2}[/tex]

I substituted that into the other equation and got

[tex]b=\frac{a^{2}}{2}[/tex]

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.

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# Homework Help: Leibniz' Integral rule

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