(This topic is also in Section 6.4 in Applied Calculus and Section 13.4 in Finite Mathematics and Applied Calculus)
To motivate the Fundamental Theorem of Calculus (see below) let us look at a cost function from the textbook (Example 5 in Section 6.1 of Applied Calculus or Section 13.1 in Finite Mathematics and Applied Calculus):
Example: The marginal cost of producing baseball caps at a production level of x caps is 4  0.001x dollars per cap. Find the total change of cost if production is increased from 100 to 200 caps.
We can solve this problem in two ways:




= 

Total Change in Cost  =  C(200)  C(100) 
=  [4(200)  0.0005(200)^{2} + K]  [4(100)  0.0005(100)^{2} + K]  
=  [780 + K]  [395 + K] = $385 
The total change of cost in going from a items to b is obtained by taking the antiderivative of the marginal cost, evaluating at x = b, evaluating at x = a, and then subtracting the answers.
But Wait! we also have this second way of doing this calculation based on the method in the previous tutorial:
Total Change in Cost  =  200 100  C'(x) dx 
This calculation says:
The total change of cost in going from a items to b is obtained by taking the definite integral of the marginal cost from a to b.
Fundamental Theorem of Calculus (FTC)
Let f be a continuous function defined on the interval [a, b] and if F is any antiderivative of f and is defined on [a, b], we have
Moreover, such an antiderivative is guaranteed to exist. In Words
Examples 
Notation: Let us redo Example 2 above, but this time introduce some notation as we go:






If we remember that definite integrals are also areas, we can use the FTC to compute areas of regions bounded by curves:
Q Consider the area in the xyplane bounded by the xaxis, the vertial lines x = 1 and x = 2, and the curve y = x^{3}:
To compute the total area (red + green) shown, we should compute:









Get me out of here! 
It is often necessary to use substitution to evaluate definite integrals. When doing so for a definite integral, remember that the limits of integration are vvalues of the variable of integraion. For instance, in
2 1 
6 
  x 
dx 
the variable of integration is x, and so the limits (1 and 2) are values of x as well. When we change variables, we should change these values to the corresponing values of the new variable as we now show:
Using the FTC with Substitution
When you change variables from, say, x to u, you need to remember that the limits in the given definite integral are limits of x, and not u. So it is a good idea to change the limits to values of u, as we see in the following example: Example 
You now have several options