The Fundamental Theorems of Calculus

The Fundamental Theorem is what we've been working up to all year. It uses derivatives, anti derivatives, and integrals to find the area beneath a graph. It states:
If f is a continuous function on the closed interval [a,b] and F is the anti derivative of f, then:

This means, in cases where we cannot find the integral of the function easily, we can use the anti derivative between two points and end up with the same answer.
The Fundamental theorem leads into the Mean Value theorem for integrals. This is another method which can be used to find the integral of a function. On any graph with the domain [a,b] there are points on the y axis, [f(a),f(b)] There must be a point between them on the graph, called f(c). Using f(c) and the difference between b and a to make a rectangle, we are able to find the integral of the function once again. There is an image of my personal notes, the important part is the graph that shows where f(c) would go.  and  are the two important functions we use from this theorem. f(c) is found by averaging f(a) and f(b).
We also covered the Second Fundamental Theorem of Calculus on Friday. The second theorem involves taking the derivative of a function defined by an integral. The format of the functions change in this theorem:
 the integral is now an equation, and we have two variables. This theorem focuses on the x in the equation. We can use this function to find the derivative of f(x).
The notes below are from class:

They explain how the second theorem is used using a sine equation for an example. We will generally not be concerned about the second variable, as it only changes the line on the graph and not what we are looking to find. The equation wants to find f(2π) so: 

This equation now looks like something we are more used to. The equation is solved like a normal integral, and the answer is 0.
In the second theorem, the first number in the integral, the "a" value, must always be a number. The second one must always be a single variable.
In class, we also found out what to do when the upper bound is not a single x. You need to find the derivative of the function, and use the first theorem to finish solving.

The last thing we looked at was what to do if the bounds of the function were two variables, rather than a number and a variable.

We can use a constant, called a in the notes, to divide the two sides of the bounds. Then, since a constant is needed in the bottom, one of the functions can be flipped, so they both run from a to x. The flipped function only needs to be multiplied by -1 and the two are added back together to find an answer for the specific problem.