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Problem 15 Easy Difficulty

Use the transformation $x=u / v, y=u v$ to evaluate the integral sum
$$\int_{1}^{2} \int_{1 / y}^{y}\left(x^{2}+y^{2}\right) d x d y+\int_{2}^{4} \int_{y / 4}^{4 / y}\left(x^{2}+y^{2}\right) d x d y$$

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Video Transcript

{'transcript': "were given a transformation and an integral and were asked to use this transformation to evaluate this integral. The transformation is X equals you over V. Why equals u V and the integral is actually the integral some integral from 1 to 2 integral from one over. Why toe Why of X squared plus y squared dx dy y plus the integral from 2 to 4 integral from wide or 4 to 4 over Why of X squared plus y squared dx dy y. So, first we will right u and V in terms of x and y so solving these systems for you and be and then we'll also find a Jacoby in of this transformation. And then we will write the boundaries of our regions in the X Y plane as the boundaries of our regions in the UV plane. And finally you're right are integral some over integral over the x y plane of as an integral some of integral in U and V because X is equal to you. Overbey Why is equal to U V? It follows that's these squared is equal to y over x, and that you squared is equal to X y the Jacoby in of her transformation. He x y do u V is the determinant of the matrix one over v negative. You over b squared the you in evaluating we get to you over V now the boundaries of our integral zin the X y well for our first into grow. This includes y equals X, and we also have Why equals four X. We also have for the other integral x times y equals one mhm and x times y was four. Now the first of these equations plugging in x and y in terms of U and V and solving we get V equals one solving the second this is V equals two solving the third This is U equals one and the fourth is U equals two and therefore are integral some integral from 1 to 2 integral from one of her. Why to why of X squared plus y squared dx dy y plus the integral from 2 to 4. And to grow from for over. Why sorry? Why over 4 to 4 over Why of x squared plus y squared dx dy y rewriting this This is now an integral from V equals one to be equals two integral from U equals 12 u equals two and we see this actually collapses into a single integral of the inter Grand X squared plus y squared becomes you square to Rive e squared plus u squared V squared times the Jacoby in which was to you over v an absolute value because both you and your positive doesn't matter and this is d UDV and simplifying. You get the end to grow from 1 to 2 and to go from 1 to 2 of this is to you cubed over B cube. Plus, this is to you cubed over the KUTV. It's very not to uncover v to you cute times v and we see here we can use for beans. The're, um right. This is a product of integral. So he had two times the integral from 1 to 2 of you cubed do you times the integral from 1 to 2 of one over v cubed plus the TV. So evaluating we get two times 1/4 times to to the 4th 16 minus one times taking anti derivatives. This is negative And then I've been negative too. So negative one over to thes squared. Plus one half B squared from 1 to 2. And this simplifies to 225 over 16, and this is our answer."}

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