We've just written some notation that says the exact area between 1 and 4 under the curve f of x and above the x-axis. In order to actually do anything really productive with this, we have to turn to the second fundamental theorem of calculus, sometimes called part 2 of the fundamental theorem of calculus, which tells us that if f has an antiderivative, so if we have the antiderivative of f, so f of x is derivative of some function capital F of x, or another way of saying it is capital F of x is the antiderivative of lowercase f of x, then I can evaluate this thing, and we do a whole video on conceptually understanding why this makes sense, we can evaluate this by evaluating the antiderivative of f or an antiderivative of f at 4 and from that subtract the antiderivative evaluated at 1. So let's do it for this particular case right over here. So we are taking, I'll just rewrite this statement, instead of writing f of x I'll write x squared, so the definite integral from 1 to 4 of x squared dx, well, we're just going to have to figure out what the antiderivative is. So if f of x is equal to x squared, what is capital F of x equal to? What is the antiderivative? Well, you might remember from your power rules that if you take the derivative with respect to x of x to the third, you are going to get 3x squared, which is pretty darn close to x squared except for this factor of 3, so let's divide both sides by 3.