There's this question in my calculus book that goes something like this:
The derivative of the area of a circle with respect to its radius is equal to the circle's circumference ($dA/dr = 2 \pi r$). Give a geometric explanation of why this is the case.
To me this is really obvious, but I find it hard to put into words. If you increased the radius of the circle by putting your finger inside it and pushing the edge outward, then you would have to push it around the whole circumference of the circle. Heh. I don't know.
Best Answer
It's because the area of a circle is really the sum of infinitely many circles' circumferences, which is the integral: $$ \int^r_0 \! 2\pi{}t \, dt $$
See http://betterexplained.com/articles/a-gentle-introduction-to-learning-calculus/ for a nice explanation