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Use a double integral to compute the area of the region bounded by the curves. $$y=x^{2}, y=x+2$$

$\frac{9}{2}$

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Johns Hopkins University

Missouri State University

Harvey Mudd College

University of Nottingham

Okay, so, in order do you figure out the area between these two functions? So I'm going to start by setting the functions equal so that I could be able to see their the intercept. So with that, I'm gonna have X squared X squared equal to X plus two. So I'm gonna bring everything to the left hand side. So that will give me X squared minus X minus two is equal to zero. From here, I can factor to figure out my, um, X value. So here I'll have X minus two times X plus one. So this will give me X equal to an X equal to negative one. So this shows me that this is negative blanc, and this is to okay, so keeping that in mind. So now we want to find that aereo's. So we're gonna take the top most function and subtract, um, the bottom most function and then put that under the integral so area is going to be equal to the integral of the top most function, which is gonna be exposed to and then minus the bottom most function, which is X squared and then times, DX or X range from negative born to chew as we found out here. So then, from here, I'm gonna just rewrite my integral so a is going to be equal to the integral from negative 1 to 2 of negative X squared plus X plus two times d x From here, I'm just gonna now take the anti derivative of each term. So the anti derivative off negative X squared is gonna be negative X cubed over three and then the anti derivative of X is gonna be X squared over two. Then the anti door of do of two is just gonna be two X and then we're gonna evaluate this from negative 1 to 2. So first, we're gonna plug in to so we have negative two cubed over three plus two squared what were too plus two times two and even now subtracts and I were gonna put in negative blanc. So negative negative one huge over three plus negative one squared over two plus two times negative one. So and with that, we will get a result of 9/2 or nine Hafs, which is equal to 4.5, and that's gonna be area of the shaded region