 So let's introduce a useful approach to integration known as changing directions. Our ability to do so rests on the geometry of the integral. In particular, a definite integral corresponds to the area of a specific geometric region. It's useful to keep this in mind because this means that any method that we have of finding the area of the region can be used to find the value of the integral. For example, suppose I want to find the definite integral from 0 to 4 of square root 16 minus x squared. This integral corresponds to the area of the region that's under y equals square root 16 minus x squared above the x-axis from x equals 0 to x equals 4. And this region looks like, and in fact, this corresponds to one quadrant of a circle with radius 4. So if we can calculate the area of this quadrant of a circle, then we can calculate the value of the integral. But it's a circle, and we know how to find the area of a circle. So let's think about this. When we set up a definite integral to find the area of a region, we look around to see if there are any bands of rogue mathematicians we have to impress, and since there aren't, we graph the region, we draw a representative rectangle, we find the height and width, and then we sum the areas of these representative rectangles from the beginning to the end of the region. And here's an important insight. The representative rectangles can go in any direction. We've been drawing them vertically, but there's no reason they can't go horizontally. For example, suppose we want to find the area under the graph of y equals log x from x equals 1 to x equals 4. So we'll graph our region, and then we'll draw our representative rectangle. Now the height of this representative rectangle is y. The width is a small portion of the x-axis, we'll call that dx. The interval starts at x equals 1, and the interval ends at x equals 4. And so we can sum up these representative rectangles whose areas are y dx from x equals 1 to x equals 4. At this point, we emphasize an important idea. The only permitted variable is that of the differential. In this case, our differential variable is x, and we're okay for this upper limit, x equals 4, and this lower limit, x equals 1, but y is not the permitted variable. Fortunately, we know that y equals log of x, so instead of writing y, which I'm not allowed to do, I can write log x, and now we can find the area by evaluating this integral. Except we can't yet evaluate this integral. We don't know how to find the antiderivative of log x. At least not yet. That'll come a little later. In a kind and gentle universe, we wouldn't have to worry about problems we couldn't solve. But we don't live in that universe. Even if we can't solve a problem, the problem still exists. So we might see if there's another way of solving this problem. So what else can we do? What if we draw our representative rectangles horizontally? Since they always end at x equals 4, their length is 4 minus x, and their width is this tiny portion of the y-axis, dy, and so their area is width times length, 4 minus x, dy, and we're still summing them up from x equals 1 to x equals 4. But remember, the only permitted variable is that of the differential. So if I'm to have any hope of evaluating this integral, everything has to be in terms of y. So again, I know that y equals log of x, and so that tells me x is e to the y. And I also need to change my limits of integration. At x equals 1, y is the log of 1, or 0, and at x equals 4, y is log 4. And so my limits of integration change, and my function changes, and that's good because this 4 minus e to the y is a function I can anti-differentiate easily. So I'll find the anti-derivative, and I'll evaluate the function at log 4 and subtract the function at 0. And after all the dust settles, get my answer for the value of the integral, which is also the area of the region.