 Hello and welcome to a screencast today about finding the area between two curves. Alright, so today we are going to determine the area of the finite region bounded by y equals x squared and y equals the squared of x. Okay, so I'm a visual person so I prefer to see the picture of this first and then from there we're going to need to figure out our intersection points and our certain slices and all that good stuff. Okay, so y equals x squared, hopefully you remember that that looks like just your basic old parabola and that's not a bad drawing there. And then the square root of x, let me draw that in a little bit different color as we pick red. That's half of a sideways parabola and it's going to go something like that and I tried to make it, I'm emphasizing these graphs by the way. So the area that we are interested in then is this kind of football, a teardrop like shape there. Okay, fantastic. So now we know what our region looks like, we need to figure out our intersection points. Okay, so you can either do this graphically with your calculator, maybe you can just look at these two functions and know where they cross, but I'm still going to walk you through the algebra of it as well. Okay, so we know that two functions intersect when they are equal. So let's go ahead and set these two functions equal to each other. So we've got x squared is the square root of x, then we've got to solve for x. And again, maybe you can look at this and know the values that are going to work, maybe not. So let's go back and review some algebra then here. So how do we undo a square root? Well, we need to square both sides. So we've got x to the fourth equals x. Let's swing that x over, x to the fourth minus x is zero. Factor an x out, so we have x times x cubed minus one is zero. Set each factor equal to zero. So we've got x is zero, we have x cubed minus one is zero, which means x cubed is one, which means x is one. Okay, so we have our two intersection points. And like I said, maybe this was a total waste of your time because you already knew what they were. But that's okay, we're going to be doing more difficult functions where that won't be the case. So let me blow up this area just a little bit for us, just so I can maybe do some better slices in here. So here is our parabola and pretend like that actually hit the origin. And then let's go ahead and draw in our square root a little bit more obvious here. So we decided our two intersection points were here at zero and then here at one. Okay, so now you have to make the choice. What kind of slices do you want to do with your area? Do you want to do vertical slices that are going to go up and down and are going to be parallel to the y-axis? Or would it make sense to do horizontal slices? Now how you can determine which way you want to do it? Well, sometimes like in this particular one, we have a choice. It doesn't matter which way we do it because if we do a vertical slice, no matter what part of the region we're in, the top of our slice is going to hit our top curve and the bottom of our slice is going to hit the bottom curve. Okay, and if we were to do horizontal slices, the right side of the rectangle would hit the right curve and the left side would hit the left curve. Okay, and in this case, you know, it may again seem a little bit crazy, but sometimes you have to do one over the other. In this case, we have a choice. So when I have a choice, I'm always going to do vertical slices. Okay, so let's go ahead and put in a slice. So there we go. I'm pretending that's actually a rectangle. Okay, so then we have to figure out what's the width of our slice and then what's the height of our slice? Well, the width and I just made this totally arbitrary. If you think about it, this can kind of trail down to your x-axis. So that width has got to be delta x. The height of this rectangle, so I'm going from my top curve down to my bottom curve. So my top curve in this case is the square root of x and my bottom curve is x squared. So the height of this rectangle is going to be the square root of x minus x squared. Okay, fabulous. So now that we have our beautiful drawing labeled, now that I've made a big mess of it all, but that's okay. Let's go ahead and do our integral. So we've got the integral from zero to one because those were our two endpoints. And again, you want to put the smaller number on the bottom and the bigger number on the top. And then we're going to do the height of our rectangle. So in this case, that's the square root of x minus x squared. And then we're going to multiply that by the width of our rectangle. But in that case, we're just going to call it dx. All right, fantastic. So now we have an integral that we want to do. I'm going to go ahead and rewrite this first piece as x to the 1 half just so I remember. And then this is a fairly straightforward integral to do. It's just different. So we can just integrate each piece separately and then throw those endpoints in. So x to the 1 half is a power. So let me go ahead and raise that by one. So that's going to give me x to the 3 halves. And then I'm going to multiply that by 2 thirds or divide by 3 halves either way. And then minus the anti-derivative of x squared is going to be, again, raised that by one and then multiplied by the reciprocal. So 1 third x cubed. And I'm going to evaluate that from zero to one. Okay, throw in my top endpoint. Throw in my bottom endpoint. And hopefully the top minus bottom is something familiar just like what we did up above. Okay, so there's my top value minus. Throw in that zero, which isn't going to give us anything on our answer, but still it's good to be consistent about these things because sometimes throwing zero in won't give you zero. Okay, so that piece wipes out and then let's see. So here we're going to get 2 thirds minus 1 third or an answer of 1 third. Fantastic. Thank you for watching.