## The Fundamental Theorem of Calculus, Part II (Practical Part) |
HELP |

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.

- If you haven't done so already, get familiar with the Fundamental Theorem of Calculus
(theoretical part) that comes before this. In particular ...
- How is the area function `A(x)` defined?
- 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?

- 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?