The Second Fundamental Theorem of Calculus

The Mean Value and Average Value Theorem For Integrals

 Let f be continuous on [a,b], then there is a c in [a,b] such that We define the average value of f(x) between a and b as

 Definition of the Average Value Example

The average value of

y = sin x

between x = 0 and x = p is The Second Fundamental Theorem of Calculus

 Let f be continuous on [a,b] then Eample Example: Solution

This is not in the form where second fundamental theorem of calculus can be applied because of the x2.  We use the chain rule so that we can apply the second fundamental theorem of calculus.  Example

Find the derivative of Solution

Here, the "x" appears on both limits.  We use two properties of integrals to write this integral as a difference of two integrals. The first integral can now be differentiated using the second fundamental theorem of calculus, The second integral can be differentiated using the chain rule as in the last example. so that Putting these results together gives the derivative of

2 sin(4x2) - sin(x2)