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The mean value theorem for integrals (not to be confused with the mean value theorem for derivatives) is used to find the average value of a function over a given interval.
To differentiate this from the Mean Value Theorem for derivatives.
The Mean Value Theorem for Integrals uses the average value of the function and guarantees that if the function is continuous on the interval then the average value is equivalent to the output of the function at at least 1 point.
The Mean Value Theorem for Derivatives uses the average rate of change (slope fomula), and guarantees for differentiable functions that the that instantaneous slope is equal to the average slope at at least 1 point.
The basic form for the MVT is the area divided by the width as shown below:
With Calculus plugged in we get the equation shown below.
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