The Fundamental Theorem of Calculus, Part II (Practical Part)

The Fundamental Theorem of Calculus, Part II goes like this: Suppose `F(x)` is an antiderivative of `f(x)`. Then

[`int_a^b f(x) dx = F(b) - F(a).`]

This might be considered the "practical" part of the FTC, because it allows us to actually compute the area between the graph and the `x`-axis. In this exploration we'll try to see why FTC part II is true.

Drag the blue points `a` and `b` to make things happen.

Explore - A Proof of FTC Part II

If you haven't done so already, get familiar with the Fundamental Theorem of Calculus (theoretical part) that comes before this. In particular ...
1. How is the area function `A(x)` defined?
2. How do you know that `A(x)` is an antiderivative of `f(x)`?
Does `A(b)` equal the integral of `f(x)` between `x = a` and `x = b`?
What is the value of `A(a)`? Why?
Justify: `int_a^b f(x) dx = A(b) - A(a)`.
Justify: If `F(x)` is an antiderivative of `f(x)`, then `F(x) = A(x) + c` for some constant `c`.
Justify: `F(b) - F(a) = A(b) - A(a)`.
Now put it all together, and you have a proof of FTC, part II, right?

Examples

Let `f(x) = x^2`. What is the area under `y = x^2`, above the `x`-axis, and between `x = 0` and `x = 1`?
- Choose an antiderivative (any antiderivative!) of `f(x) = x^2` and call it `F(x)`.
- Compute `F(1) - F(0)`. Does your answer agree with the applet above?
Let `f(x) = sin(x)` and repeat the above procedure to find the area under one "hill" of the sine curve.
Compute `int_(-1)^1 e^x dx`. What is the easiest `F(x)` to choose? Will it be the same as the one graphed in the right applet?

For Further Thought

We officially compute an integral `int_a^b f(x) dx` by using Riemann sums. But now, FTC part II gives us a much easier way to compute definite integrals - as long as we can find an antiderivative. Why is `int_0^1 2xe^(x^2) dx` easy to compute using FTC part II, but `int_0^1 e^(x^2) dx` is not?