Okay, so here we have a region to find by X equals the absolute value of why and X equals two minus y squared. And we want to find the area of this region both with respect, X and disrespectful. I were to do it both ways. We want to make sure that her answer matches at the end. So we know our general form. Yes, Our area is equal to the integral from A to B of F minus G. We're gonna do it for FX, G of X and F of Weijia. A few notes We know that our X equals absolute value of why is actually essentially to separate equations. We have our X equals y in the positive quadrant and we have an X equals negative. Why in the negative quadrant, in terms of X, they're basically the same thing we have like was X And why equals negative X Our parabola X equals two minus y squared. If we want it in terms of X, we get why equals plus or minus the absolute value of two minus X. So now we have our equations in terms of X and Y we're gonna need dollar various points. So we're gonna have potentially these four points. We have the two intersection points and we have to ex intersection point. No, our origin is easy. X equals wide 00 So that's that point. Our parabola is pretty straightforward. Uh, it's gonna be 20 The intersection points. We'll find this one. We have X equals y and X equals two minus y squared. If we set those equal to each other, we get Why equals two minus y squared. You can rearrange it into quadratic equation, and it factors into why. Plus two, Why minus one. So we get why equals negative two and one. The negative two is off our graph. It's not even relevant to the region we're looking at. So are one is important. Plug it in and we get the 0.11 Now we could do that again for the bottom, but we know that both of these air symmetrical around the X axis, so we know that they're going to have the same X value and opposing y values. So we know this is simply one negative one, and that gets us all our our parts. We have the boundaries for the intervals with our various intersection points on. We have all of our equations in terms of X Ally, Let's get started. Let's find our area in terms of X first. So in terms of X, we're going to use a vertical line, Teoh, to check our regions, we're gonna have to regions, you know, divided by the line at X equals one. So our first region is our absolute value and since it's symmetrical, we're just going to use our positive, are positive line and multiply it all by two. So we have to times the Inter girl from zero toe one of X. Yes, plus, her second region is just the parabola. So again, we're just gonna use the positive value and multiply the whole thing by two so it will be one to to and radical. Tu minus X dx are integral for radical two minus X is going to require some use substitution or you is gonna be two minus X and D you is going to be negative. DX then when you resolve that are anti derivative is gonna be two times the quantity of X squared over two from 0 to 1 minus to times two times tu minus X to the three halves all over three from 1 to 2. When we substitute, we're going to end up with two times the quantity of 1/2 minus zero minus two times the quantity of 0/3, minus 2/3. And then that should all simplify down to 7/3. Right? And now, with respect to why, with our horizontal line test, we see that our we're gonna have to regions separated by the X axis. So our first region, our lower region, is gonna go from negative one 20 And with a horizontal line test, the right most equation is our parabolas. That's gonna be our f of X or f of why rather tu minus y squared that will be minus negative. Why or plus Why? Because our g of X is negative. Y do I plus integral from 0 to 1 Once again, our parabola is our f of why and this time RG of y is simply why so it's minus or g a y de y. This doesn't require any use institution. We can just get our anti derivatives right away. So we end up with two. Why minus why cubed over three plus why squared over two from negative 120 plus to I minus y cubed over three minus y squared over to from 0 to 1. Our substitution gets us zero minus the quantity of negative two plus 1/3 plus 1/2 plus the quantity of two minus 1/3 minus 1/2 minus zero. And again, when you simplify everything down, we should come out with 7/3. So the area of this region is 7/3 and that's it. We're done.