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\begin{equation}

\begin{array}{c}{\text { Use the transformation } x=u^{2}-v^{2}, y=2 u v \text { to evaluate the }} \\ {\int_{0}^{1} \int_{0}^{2 \sqrt{1-x}} \sqrt{x^{2}+y^{2}} d y d x} \\ {\text { (Hint: Show that the image of the triangular region } G \text { with vertices }} \\ {(0,0),(1,0),(1,1) \text { in the } u v-\text { plane is the region of integration } R \text { in }} \\ {\text { the } x y \text { -plane defined by the limits of integration.) }}\end{array}

\end{equation}

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So I started to write out the relation between that's why you'll be so X equals two. You squirt guns. We swear y calls to to u v So that gives us the Jacoby in J F. U V equals two to you, honest to V to be into you. So the answer is four times use worthless we scored. And the second step is to define the boundary Oh, on over g region. So by g region, we mean the region transport from our on the plane too the, you know, be clean plane. The question y equals two to him Swirled oneness x movie transform 22 You, we swear equals 24 terms one month's use word minus re square close B squared And this gives us you equals two plus like it warm. And why zero will be transformed to Ewing, Cyril or B equals zero. Execute zero emissions. Fun to use be or you close read with zero. So that's all the hungry for over you and me here 123 And this fun step is the things a girl. So we have this already from 0 to 10 to two thumbs swirled woman's ex Screw the X squared plus y squared the XT Iwai! So this is gonna be 0 to 10 to you swear ou that the so inside we have use worms with words squirt close to you the score So this actually use squared plus B squared times itself. So it's you squared, plus b squared, squared. And don't forget we have that Jacoby here a silken times four times you squared plus b squared and this is DVD You simplified into grins. We're going in the void 0102 You four temps you squared plus B square and this squared DVD you and said, Hi, symmetrical. It's a girl. And finally we're gonna have 112 or 15 times you've been six over six and that this is 12 to the range. So the final answer will be 56 over 45

University of Illinois at Urbana-Champaign