The Fundamental Theory of Calculus part 2 (part ii)
OK, get ready for some Calculus-Fu!
We have now said that rather than taking pesky limits of Riemann sums to calculate areas under curves (ie. definite integrals), all we need is to find an antiderivative of the function that we are looking at.
As a reminder, to calculate the definite integral of a continuous function, we have:
where is any antiderivative of
Remember that to calculate the area under the curve of from, let’s say 2 to 5, we had to write:
And at that point we had barely even started because we still had to actually evaluate this sum, which is a hell of a calculation…then we have to calculate the limit. What a pain.
Now, we are told that all we have to do is to find any antiderivative of and we are basically done.
Can we find a function which, when we take its derivative gives us ?…