Evaluate the integral by making an appropriate change of variables. double integral x−2y3x−ydA, where R is the parallelogram enclosed by the lines x−2y=0, x−2y=4, 3x−y=1 and 3x−y=8

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Marlene Broomfield · Accepted Answer

Step 1 1) Let&#039;s start by substituting u=x−2y and v=3x−y. From there we get y=v−3u5 x=2v−u5 2) R is a rectangle defined by 0≤u≤4 and 1≤v≤8. We will need this later as bounds for integration. 3) Now we are ready to find the Jacobian. ∂(x,y)∂(u,v)=|−1/52/5−3/51/5| =15 4) Now we can evaluate the integral. ∫∫Rx−2y3x−ydA=∫04∫18uv|15|dvdu 5) Notice that we can separate the integrals. Thus getting the following. =15∫04udu∫181vdv =15⋅[12u2]04⋅[ln⁡v]18 =85ln⁡8

Mary Ramirez · Accepted Answer

Let us substitute u=x−2y, v=3x−y.    Hence x=2v−u5, y=v−3u5. Then Jacobian    J=≤ft|∂(x,y)∂(u,v)right|=xuyv−xvyu=15.    Now ∫∫Rx−2y3x−ydA    =∫∫Ruv≤ft|∂(x,y)∂(u,v)right|dudv    =∫u=04∫v=18uv15dudv    [u22]04[ln⁡v]1815    =85ln⁡8    Answer    ∫∫Rx−2y3x−ydA=85ln⁡8

Evaluate the integral by making an appropriate change of variables.

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