The
Fundamental Theorem of Calculus
Suppose you start with a function f, and then integrate it from
a constant lower limit to a variable upper limit. The result should
depend on the variable in the upper limit, and so the result is a new function.
If you then differentiate this new function, what is the result?
The mathematics below illustrates what happens. This remarkable result
is known as the First Fundamental Theorem of Calculus.
Change the formula for the function f, and see how the last line (the
derivative of the integral) changes. Also try changing the constant
in the lower limit of the integral, to see how this affects the final result.
If the mathematics becomes hopelessly messed up, just click the Reload
button on your browser.