Change of Variables Formulaд(х, у)SSof(x, y) d x d y = SLf(x(u, v), y(u, v))|dudva(u,v)6. Evaluate the integral [xy dA where D is the region in the first quadrant bounded by the equationsy = x, y = 4 x, x y = 1, and x y = 4. HINT: Consider the change of variables u = x y and v = y.

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Asked Dec 6, 2019
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Evaluate the integral ∫∫ D x y dA where D is the region in the first quadrant bounded by the equations y=x, y=4x, x y=1, and x y=4. HINT: Consider the change of variables u=x y and v=y.

 

Change of Variables Formula
д(х, у)
SSof(x, y) d x d y = SLf(x(u, v), y(u, v))
|dudv
a(u,v)
6. Evaluate the integral [xy dA where D is the region in the first quadrant bounded by the equations
y = x, y = 4 x, x y = 1, and x y = 4. HINT: Consider the change of variables u = x y and v = y.
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Change of Variables Formula д(х, у) SSof(x, y) d x d y = SLf(x(u, v), y(u, v)) |dudv a(u,v) 6. Evaluate the integral [xy dA where D is the region in the first quadrant bounded by the equations y = x, y = 4 x, x y = 1, and x y = 4. HINT: Consider the change of variables u = x y and v = y.

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Step 1

Evaluate integral by change of variables.

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Given: If xydA Change of variable are: u=xy and v=y. since, y=v. u=xv и х

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Step 2

Region bounded by the equation are:

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y=x, y=4x, xy=1,xy=4. From the equation,xy=1,xy = 4. u varies from 1 to 4. From the equation, 2=1,2=4 х х →=1,- :=1 u/v и/v =1,- и =u,v² =4u =v=Ju,v=J4u v varies from u and 2 Ju.

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Step 3

Find Jacobian of this ...

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|x ôx Ô(x,y) _ ĉu ôv бу дy |аи ду д(и, у) и и v2 и v' ,2 v2

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