OK, so up to now we can’t actually use the FTC (Fundamental Theorem of Calculus) to calculate any areas. That will come from the FTC part 2.
For now, let’s take some examples and see what the FTC is saying. I’ll restate it here:
The Fundamental Theorem of Calculus, part 1
If is continuous on then the function defined by:
is continuous on and differentiable on , and .
Let’s look at some examples. We’re going to take an example that we can calculate using a Riemann sum. Let’s choose .
If we integrate this from to some point – ie. calculate the area under the curve, we get:
Make sure that you can indeed get this by calculating the Riemann sum.
So, what does the FTC part 1 tell us? It says that if we take the derivative of this area, with respect to the upper limit, , then we get back .…