 Hello and welcome to another screencast about finding the area between two curves. Alright, so today we're going to look at the area of the finite region bounded by x equals the quantity of y minus 2 squared and x plus y squared equals 10. Now I've already used geograba to graph these two functions, and that's only because my drawings would not look nearly this accurate. So we gotta figure out which one is which. So looking here at the red one, you notice that it intersects our x axis at 10. So with these two functions, if we were to plug in y equals zero, it should give us an x value of 10. Okay, so this one here is obviously gotta give us the red one. Then the black one here kind of play in the same game. But in this case, if we plug in x equals zero, it's gonna give us a y value of 2, right? So that one makes sense here for the black one. Okay, so now that we've got those, we know that we're gonna need to figure out the area between these two curves. And we've got an intersection point here, and we've got an intersection point here. And we're gonna be going between these two intersection points. So then the area that I wanna shade, let me grab my highlighter here, is gonna be all of this stuff in here. And pretend like I'm actually hitting both curves the whole way. Okay, so that's gonna be the area then that I wanna figure out. So we need to figure out our endpoints, or where we are intersecting at. And again, because I've graphed this so nicely, I could do it graphically. But I would like to make sure you guys can see how the algebra works out as well. Okay, so we wanna set these two functions equal to each other. And you notice that the function here is already solved for x, okay? And trying to get the y by itself on this one, that's not gonna be very pretty. Okay, so I'm looking at my second function over here then, and I wanna go ahead and solve for x on that one. So let's swing that y squared over. So x is gonna be equal to negative y squared plus 10. So now I can go ahead and set those two functions equal to each other. So I have y minus 2 squared is negative y squared plus 10. Let's go ahead and multiply this out. So I'm gonna get y squared minus 4y plus 4. Cuz remember, y minus 2 quantity squared means y minus 2 times y minus 2. Okay, and then I brought down my other side. Let's go ahead and solve this equal to 0. So let me bring these two pieces over to the other side. So I'm gonna have 2y squared minus 4y. And then I've got a positive 4, but I'm gonna be subtracting a 10. So that's gonna give me a minus 6 equals 0. And then I notice that all three of my terms have a 2 in common. So let me go ahead and factor that out. And then now I've got a polynomial left that I can go ahead and factor. So that's gonna factor into y minus 3 and y plus 1. And then now I can set each factor equal to 0. And I get my two intersection points this time in terms of y, okay? Last video that we did, I solved and we got our intersection points in terms of x, okay? But if you have x, you can always find y or vice versa. So that part doesn't really matter. But it just turns out that the way this function is set up, having this y inside of these parentheses, even over here, having that y squared, it's gonna be not very pretty to solve for. And we're gonna end up with pluses and minuses, and that's just not nice. Okay, so we got our intersection points done. We know what our region looks like. So now we have to figure out, all right, what kind of a slice is gonna make sense here? So I'm gonna start with vertical. So if I were to make a vertical slice, that means it's gonna go up and down. If I were to make it here in the middle, I'd have my top function, or the top part of my slice would hit my top function, and the bottom part would hit my bottom function. But you notice over here all the way to the left, if I were to make a slice right here, my top function hits this black function, and my bottom, the bottom, sorry, the top part of my slice hits this function, and the bottom part hits the same function, okay? That's not good. So I probably do not wanna do vertical slices. Let me go ahead and erase that. Okay, let's look at horizontal slices, though. So no matter where I'm at, so if I'm up here at the top, if I'm in here in the middle, or if I'm down here at the bottom, hopefully you guys can see that no matter where I make my slices, the one side of the slice is gonna hit one function, the other slice is gonna hit the other function. Okay, so let me go ahead and draw in one of those. Oh boy, this is gonna be a long slice. Okay, so pretend like there is my slice. Ooh, it got a little skinny. Okay, so here's my slice. Kinda looks like a snake. All right, so I need to figure out what the width of that slice is and what the height of that slice is. So the height, you'll notice, kinda goes back here to my y-axis. So that means the height of my slice is gonna be delta y. Okay, now how do I find the width of my slice? Well, if you look in your book or if you watch the quick recap, you notice that you always need to do your bigger function minus your smaller function, okay? Or in this case, you know, when we did the vertical slices, it was top minus bottom. In this case, it actually needs to be the right function minus the left function, okay? So a little bit uglier, but hey, it can be done. So the width of my slice then is going to be, let's see this y squared, it's negative y squared plus 10. And then minus my other function, which is gonna be y minus two quantity squared. So again, this is my right slice minus my, or my right function minus my left function is another way you can think about that. Okay, fantastic. This one's a little bit uglier than the last one, but yeah, that's okay, what can we do? All right, let's go ahead and set up our integral. So I've got the integral from my bottom endpoint, which in this case is negative one, to my top endpoint, which is three. And then I'm gonna go ahead and do my slice. So that's gonna be, let's see, we have negative y squared plus 10 minus y minus two squared dy. Okay, this is a big mess, but it's just gonna be a polynomial, okay? That's not hard to do. So we gotta multiply this piece out again, distribute our negative, or if you wanna, let's see, if we bring down this function, we're actually gonna have to make it negative, but that's okay, so let me go ahead and do some multiplying out here of things. Or this may be where your professor just says, oh, that's good enough, I just want you to know how to set it up. But just in case, that's not the case, let's go ahead and figure out what this is then. So we have y squared minus four y plus four dy. And then so that's gonna give us, sorry about kind of the messiness here, we have negative one to three. We're gonna end up with negative two y squared plus four y minus six. Let me go ahead and do that. And again, that's dy. This is a polynomial, so it should be fairly straightforward to integrate. So we're gonna end up with negative two thirds y cubed plus two y squared minus six y. And again, you can always go through and take the derivative to check it just to make sure you've got the right function in there. I'd hate to do all this work and then have the wrong function. Okay, let's throw in three, throw in negative one. And when we throw in that negative one, you definitely wanna make sure to have parentheses in there. So we've got negative two thirds three cubed plus two times three squared minus six times C times three. That whole, whatever that number is, minus negative two thirds times negative one cubed plus two times negative one squared minus six times negative one. That's a number. And then when you combine these two numbers together, hopefully my math is correct here, you end up with 64 over three. Fabulous. Thank you for watching.