I was talking with a student recently who told me that he always found the fact that amazing. “How is it that the area under one hump of the sine curve comes out exactly 2?” He asked me if there is an easy way to see that, or is it something you just have to discover by doing the computation.
If you’ve wondered about this too, perhaps you’ll find the following of interest.
Imagine a particle travelling counterclockwise around a unit circle with unit speed. Let’s say that it starts moving from the rightmost point of the circle to the leftmost point. (Orient so that the vertical axis is truly vertical.) It’s high noon and the sun is casting a shadow of the particle straight down on the ground. The shadow only moves in the horizontal direction. What is the shadow’s speed when the particle is at ? It’s . (Since we’re interested in distance traveled, we measure speed as a positive quantity.)
If the particle has speed 1, its shadow has speed .
The integral of speed is distance.
So the integral of the sine function over one hump is the total distance that the shadow travels, which is exactly the diameter of the circle.
In fact, with just a little tweaking, this setup also makes it clear that the indefinite integral of sine is the negative of cosine.