Leibniz' Integral rule

In summary, the conversation discusses the use of Leibniz' integral rule for differentiating under the integral sign in order to determine constants a and b for a given integral, with the goal of minimizing its value. The process involves taking the partial derivative with respect to x and setting it to zero, which results in two equations. After integrating and solving for x and b, the correct values are found. The conversation ends with a realization that the integration step was initially forgotten.
  • #1
adm_strat
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[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 [tex]\int^{1}_{0}(ax+b-x^{2})^{2} dx [/tex] is as small as possible.



Homework 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]



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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  • #2
Did you remember to integrate fx from 0 to 1 before setting the whole thing to zero?
 
  • #3
Duh, no I didn't integrate. Lol, I feel kinda dumb after that one. Thanks
 

What is Leibniz' Integral rule?

Leibniz' Integral rule, also known as the Leibniz rule or differentiation under the integral sign, is a mathematical theorem that provides a method for differentiating integrals that depend on a variable upper limit.

Who discovered Leibniz' Integral rule?

The rule was discovered by the German mathematician and philosopher, Gottfried Wilhelm Leibniz, in the late 17th century.

What is the formula for Leibniz' Integral rule?

The formula for Leibniz' Integral rule is: d/dx ∫g(x)f(x)dx = f(g(x))*g'(x) + ∫g(x)f'(x)dx

What is the significance of Leibniz' Integral rule?

Leibniz' Integral rule is an important tool in calculus and is widely used in various fields of science and engineering to solve complex integrals. It allows for the differentiation of integrals with variable upper limits, which would otherwise be difficult to solve using traditional methods.

What are some real-world applications of Leibniz' Integral rule?

Leibniz' Integral rule is used in various fields such as physics, engineering, and economics to solve problems involving optimization, probability, and differential equations. It is also used in signal processing, control theory, and statistics.

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