This page contains a Sage interact which illustrates the mean value theorem for definite integrals as well as links to a pair of complementary Javascript animations which illustrate the first fundamental theorem of calculus.
Use the slider in the interact below to morph the region under the graph of $f$ over the interval $[a,b]$ into a rectangle (also over $[a,b]$) with the same area as the original region. Given that $f$ is continuous on $[a,b]$ the (signed) height of this rectangle is necessarily between the minimum and maximum values of $f$ on $[a,b]$ and matches the height of $f$ at at least one point of $[a,b]$, as required by the mean value theorem.
Click to view complementary definite integration and differentiation animations which illustrate the first fundamental theorem of calculus.