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YZ

# Use the transformation in Exercise 1 to evaluate the integral$$\iint_{R}\left(2 x^{2}-x y-y^{2}\right) d x d y$$for the region $R$ in the first quadrant bounded by the lines $y=-2 x+4, y=-2 x+7, y=x-2,$ and $y=x+1$

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##### Lily A.

Johns Hopkins University

##### Catherine R.

Missouri State University

##### Heather Z.

Oregon State University

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### Video Transcript

So can I Would transform the integral from x and y to you. Andy, we have two x squared minus X Y No, this was squared the ex you, Roy. And what we know is you equals two x minus y and be close to him to expose What? Okay, so here are we gonna have do teams be times that you killed in the delta X Y over will you be? And now the UDV So from here we know that the decoding for that's why I would tell you it was too the next over another you that I said Would that be? Doesn't why over data you and other way over that we The neural value for this is 1/3 So now we have well third time's the region out this g So the gsc region transformed by over x What? You'll be transformation. You know we have you become sti you devi And now we're gonna have soon from over are we know that the region is naked naked to x minus four plus four and to us for seven. And why equals two X minus two? Why customer X plus one. So here's the region for, uh, Alexis. What plea, which is called, are no interest from the region to our you'll be playing, which is called G. Let's see. So for this line, um, boy, close to four months to x. I'm sorry. Seven months to us, this goes to big was four by over relation. And for why? Because performance to X, we would transform to every week with seven for X equals y. This explains to you we transform into U equals two and the four y equals X plus one. We transform, too. U equals negative one. Okay, so right now we have all the boundary for the region. G cell week handy. Bella Too integral. We're gonna have. And that could 12 to 4 over you and the 47 for V. The inter grants are you B? We have the video respectively. In this integral will be validated too. You don't know four times you squared. Let's see from like, 1 to 2. And finally answer for this really 33 or four

YZ
University of Illinois at Urbana-Champaign

#### Topics

Limits

Integrals

Multiple Integrals

##### Lily A.

Johns Hopkins University

##### Catherine R.

Missouri State University

##### Heather Z.

Oregon State University

Lectures

Join Bootcamp