 In this video, we're gonna discuss the solution for question eight on the practice midterm exam for calculus two, math 12-20. We are asked to set up the integral of the region bounded by y equals x squared and y equals x plus two. So we're looking for the area between the two curves and one thing to notice here is that we're just trying to set up the integral. We're not actually going to evaluate it whatsoever. Now when it comes to regions, areas like this, oftentimes it's useful to think of the picture, right? So if we think of like our x and our y axis right here, let's graph these things. I'll do the parabola in blue, y equals x squared. This is just a standard parabola. We get something like the following, right? And then we have the line y equals x plus two. That's a line that would have a y intercept of two. Its slope is one. So it's gonna do something like this, right? That's in green there. And so we can see that the region that we're trying to describe appears to be this region right here of interest to us is gonna be these intersection points, which we see here and here. So let's look for those real quick. So if we set x squared equal to x plus two, this is a quadratic equation, I'll set it equal to zero. So we get x squared minus x minus two equals zero. You can use the quadratic formula if you wanted to. But I bet that your professor is a nice fella who wants to give you ones that factor with integers intercepts there. Let's see if that happens. We want factors in negative two that add to negative one, not a lot of options for a prime number. So you're gonna get x minus two on x plus one. And so this is gonna give you as your intercepts x equals positive two and negative one. And based upon my drawing here, that actually seems quite feasible to a negative one there. That's great. And so as we integrate this thing, we're gonna integrate from these bounds. We're gonna go from negative two. Sorry, negative, I wrote negative two there. I didn't mean to. Aha, negative one. We're gonna integrate from negative one to two. We're gonna take the larger function, which is x plus two, minus the smaller function, which in this case is x squared, dx. Now, I do ask you to simplify this. I mean, admittedly, I would accept this as correct. But if we were to simplify it, right, combine some like terms. You're gonna end up with two plus x minus x squared dx. And that, this would give us the area between the two curves there. And so you're gonna see on this one that oftentimes we have to find the places where the two curves intersect each other. And so that's a technique you wanna practice for this test. Some comments I wanna make here is that when it comes to these definite integrals, things that I do need to see is you do need to have the correct function, that's something I'm gonna scrutinize. You need to have the correct bounds and it needs to start and stop at the right place. And also, you need to have your differential. That dx is part of the integral. Without it, you wouldn't get full credit on this question. So make sure to put your differential there.