 So now we have an idea of what we're looking for. We're looking for a definite integral. The question is how do we find it? And so that takes us to the fundamental theorem of calculus. And one of the important things about the fundamental theorem of calculus is that there's about three or four different ways of stating the fundamental theorem. They all say essentially the same thing, but they focus on different aspects of the statement. So here's one method of stating the fundamental theorem of calculus. In general, if I have this definite integral, so this is the definite integral from x equals a to b, f of x, dx, any definite integral, the value is going to be some function at b, that's our upper limit, minus the same function at a, that's our lower limit. And the connection between the two is that the derivative of our function is the original integrand. And one of the neat things about it is remember that derivative corresponds to slope of line tangent to graph. And so here's a really neat connection between the slope of line tangent to graph and the area underneath the curve. The two are very closely related, and their relationship is through the fundamental theorem. Well, one bit of notation here, we often write it this way, but frequently the notation that we use to indicate this is what's called bar notation. So this is f of x, and there's a vertical bar here where I indicate x equals a to x equals b, that's the limits up here. And this bar notation means essentially the same thing. It's the difference of the two function values. So what about that capital F function? Well again, the derivative of capital F is lowercase f, and so capital F is an anti-derivative of f of x. And that justifies the use of this symbol for the anti-derivative. It's the connection here that's important. And that's great, because it means that if I want to find the area under a curve, all I have to do is find an anti-derivative, except there's one problem. Remember that when you find an anti-derivative, there's that constant, and we have to figure out what the constant is. So how are we going to do that? Well, let's employ a tactic we often use in mathematics, science, technology, and a whole bunch of other areas. We have a problem that we need to solve. Let's put it aside and not solve it until we actually need the solution. So let's take a look at that. So I suppose I want to find the area between y equals x squared and y equals 4. We've already found this as a definite integral. So there is our expression for the area, and the fundamental theorem of calculus says if I want to evaluate the integral, I need to find a function whose derivative is 4 minus x squared. So that says that my function is the anti-derivative 4 minus x squared. So I'll go ahead and find that. And there's that constant of anti-differentiation that I have to worry about. Well, let's see if I really need to worry about that. So again, my fundamental theorem of calculus says if I want to evaluate the definite integral, find an anti-derivative evaluated at the end points and find the difference. So I need to figure out the value of this at x equals negative 2. I need to find the value of this at x equals 2. And then I need to subtract the one from the other. So let's go ahead and find those. So at 2, this expression, 4 times 2 minus 1 third to cubed plus c. That works out to be 16 thirds plus c. And at negative 2, this expression substituting x into the expression. And after all the dust settles, minus 16 thirds plus c. Here is what is often the hardest part of any of these definite integral problems. We actually have to do a computation there. So let's see, 16 thirds plus c, there's our f of 2, upper value. Minus f of negative 2, parentheses are cheap, use them. So here we're subtracting that. And again, here's where many, many, many, many mistakes occur. That minus a negative 16 thirds, that's plus 16 thirds, minus positive c, that's minus c. And a really useful thing happens. The plus c and the minus c cancel each other out. And all I'm left with is 32 over 3. So what this means is that while finding the antiderivative generally introduces this constant of anti-differentiation, it makes absolutely no difference whether we know what that value is or not. We can pick any number that we want to for that constant of anti-differentiation in this problem where we're evaluating a definite integral. It makes absolutely no difference what we choose. So we could omit it, but it's generally good style to at least include it in the write-up of the problem. If you omit it, it's like saying that it's equal to zero and there's nothing truly objectionable about it. But it's a good habit to get into that whenever you find an anti-derivative include that it has a constant of anti-differentiation.