Evaluate the integral by making an appropriate change of variables.

CMIIh 2021-09-15 Answered
Evaluate the integral by making an appropriate change of variables.

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Expert Answer

svartmaleJ
Answered 2021-09-16 Author has 7733 answers

Step 1
Evaluate the integral by making an appropriate change of variables. Step 2
Use the change of the variables in a double integral: Suppose that \(\displaystyle{T}\ \text{ is }\ {C}^{{1}}\) transformation whose Jacobian is nonzero and that T maps a region S in the uv-plane onto aregion R inthe xy - plane. Supposethat f is continuouson R and that R and S are type I ortype II plane regions. Suppose also that T is one-to-one, except perhaps on the boundary of S.
Then
\(\int \int_R f(x,y)dA=\int\int_S f(x(u,v),y(u,v)) \left|\frac{\partial(x,y)}{\partial(u,v)}\right|dudv\)
Step 3
Jacobian transformation
\(K\frac{\partial(x,y)}{\partial(u,v)}=\begin{vmatrix}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}\)
\(\displaystyle={\frac{{\partial{x}}}{{\partial{u}}}}\cdot{\frac{{\partial{y}}}{{\partial{v}}}}-{\frac{{\partial{x}}}{{\partial{v}}}}\cdot{\frac{{\partial{y}}}{{\partial{u}}}}\)

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