EJERCICIO 48

Description

Considere \(\displaystyle \iint_R\frac{(y-x)^4}{(x+y)^2}dA\), donde R es la región en el plano \(xy\) limitada por el trapecio con vértices (x = 1,y = 0), (x = 4,y = 0), (x = 0,y = 1) y (x = 0,y = 4). El cambio de variables \(u=x+y\), \(v=y-x\) define una transformación \(T(x,y)=(u,v)\) del plano \(xy\) al plano \(uv\). a) Muestre S y escriba S (en términos de u y v) usando notación de conjuntos, donde T:R→S. Use T para mostrar S en el plano \(uv\) evaluando T en los vértices. b) Evalúe el determinante \(\displaystyle\frac{\partial(u,v)}{\partial(x,y)}=\begin{vmatrix} \frac{\partial u}{\partial x}&\frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial x}&\frac{\partial v}{\partial y} \end{vmatrix} \). c) El jacobiano es \(\displaystyle \frac{\partial(x,y)}{\partial(u,v)}=\frac{1}{\frac{\partial(u,v)}{\partial(x,y)}}\), use esta condición para evaluar \(\displaystyle \iint_R\frac{(y-x)^4}{(x+y)^2}dA\).

Consider \(\displaystyle \iint_R\frac{(y-x)^4}{(x+y)^2}dA\), where R is the region in the \(xy\)-plane bounde by the trapezoid with vertices (x = 1,y = 0), (x = 4,y = 0), (x = 0,y = 1) and (x = 0,y = 4). The change of variables \(u=x+y\), \(v=y-x\) defines a transformation latex]T(x,y)=(u,v)[/latex] from the \(xy\)-plane to the \(uv\)-plane. a) Sketch S and write S (in terms of u and v) using set-builder notation, where T:R→S. Use T to help you sketch S in the \(uv\)-plane by evaluating T at the vertices. b) Evaluate the determinant \(\displaystyle\frac{\partial(u,v)}{\partial(x,y)}=\begin{vmatrix} \frac{\partial u}{\partial x}&\frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial x}&\frac{\partial v}{\partial y} \end{vmatrix} \) c) The jacobian is \(\displaystyle \frac{\partial(x,y)}{\partial(u,v)}=\frac{1}{\frac{\partial(u,v)}{\partial(x,y)}}\), use this fact to help evaluate \(\displaystyle \iint_R\frac{(y-x)^4}{(x+y)^2}dA\).