Leibnitz Rule For an Integral.

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In summary, the given equation shows how to find a particular integral in a stat book by using the limit definition of the derivative. By adding zero in a creative way within the limit expression, the integral can be rewritten to include other terms that can then be evaluated using the limit definition. This allows for the computation of the particular integral without explicitly knowing the function being integrated.
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X89codered89X
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I found a particular integral in my stat book.

[itex] \frac{d}{ d\theta}\int^{b(\theta)}_{a(\theta)}f(\theta,t)dt =

\int^{b(\theta)}_{a( \theta)}\frac{ \partial}{ \partial \theta}f( \theta ,t)dt +

f( \theta, b( \theta)) \frac {\partial b(\theta)}{ \partial \theta} -

f(\theta, a(\theta))\frac{ \partial a(\theta)}{\partial \theta} [/itex]

Why is this the case? Why is it not...

[itex] \int^{b(\theta)}_{a(\theta)}f(\theta,t)dt =

\int^{b(\theta)}_{a( \theta)}\frac{ \partial}{ \partial \theta}f( \theta ,t)dt +

\frac{d}{ d\theta} [F( \theta, b( \theta)) - F(\theta, a(\theta))]

[/itex]
EDITED: Fixing LaTeX, as per usual. Sorry Folks.
 
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1. Set up the limit definition of the derivative.
2. Now, you have to add zero in a creative way within the limit expression.
3. For example:
[tex]\int_{a(\theta+h)}^{b(\theta+h)}f(\theta+h,t)dt=\int_{a(\theta+h)}^{b(\theta+h)}f(\theta+h,t)dt-\int_{a(\theta+h)}^{b(\theta)}f(\theta+h,t)dt+\int_{a(\theta+h)}^{b(\theta)}f(\theta+h,t)dt=\int_{b(\theta)}^{b(\theta+h)}f(\theta+h,t)dt+\int_{a(\theta+h)}^{b(\theta)}f(\theta+h,t)dt[/tex]
The first integral is readily seen to converge (when divided by h and letting h go to zero) to the second term in your first line's RHS.
With the two remainding integrals within your limiting expression, add another creative zero to get the other two terms in your first line's RHS
 
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What is the Leibnitz Rule for an Integral?

The Leibnitz Rule for an Integral is a formula used in calculus to find the derivative of a definite integral. It allows for the differentiation of a function that is defined as an integral with variable limits of integration.

What is the formula for the Leibnitz Rule?

The formula for the Leibnitz Rule is:

d/dx ∫a(x) to b(x) f(t)dt = f(b(x)) * b'(x) - f(a(x)) * a'(x) + ∫a(x) to b(x) f'(t)dt

where a(x) and b(x) are functions of x and f(t) is the integrand.

What is the significance of the Leibnitz Rule?

The Leibnitz Rule is significant in calculus because it allows for the differentiation of a definite integral, which is a fundamental operation in calculus. This rule can be used to solve a variety of problems in physics, engineering, and other fields that involve the calculation of derivatives of integrals.

How is the Leibnitz Rule applied?

The Leibnitz Rule is applied by using the formula to find the derivative of a definite integral. The first step is to identify the functions a(x) and b(x) and the integrand f(t). Then, plug these values into the formula and simplify the expression. Finally, integrate the simplified expression to obtain the final answer.

What are some common mistakes made when using the Leibnitz Rule?

Some common mistakes made when using the Leibnitz Rule include incorrect identification of the functions a(x) and b(x), incorrect application of the formula, and not simplifying the expression before integrating it. It is important to double-check the inputs and steps when using this rule to avoid errors in the final answer.

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