EJERCICIO 48

Descripción

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 $\displaystyle \iint_R\frac{(y-x)^4}{(x+y)^2}dA$.