Exercise · ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-009 · ★★★ Advanced
Let \(R\) be the triangle with vertices \((0,0),(1,0),(0,1)\). With \(u=y-x,\ v=x+y\), compute \(\iint_R e^{(y-x)/(y+x)}\,dA\).
Choose one option.
A \(\tfrac{e-e^{-1}}{2}\) B \(e-e^{-1}\) C \(\tfrac{e-1}{4}\) D \(\tfrac{e-e^{-1}}{4}\) Show answer D. \(\tfrac{e-e^{-1}}{4}\)← Back to the question
Explanation \(R\) becomes \(0\le v\le1,\ -v\le u\le v\), with \(|J|=\tfrac12\). Then \(\int_0^1\!\int_{-v}^{v}\tfrac12e^{u/v}\,du\,dv=\int_0^1\tfrac12 v(e-e^{-1})\,dv=\tfrac{e-e^{-1}}{4}\).
A change of variables turns the integral of a function \(f\) over a region \(R\) of the \(xy\)-plane into one over a region \(S\) of the \(uv\)-plane, correcting the area with the Jacobian \(J\) of the transformation.
\[\begin{aligned}\iint_R f\,dx\,dy&=\iint_S f(x(u,v),y(u,v))\,|J|\,du\,dv\\J&=\frac{\partial(x,y)}{\partial(u,v)}=\begin{vmatrix}x_u&x_v\\y_u&y_v\end{vmatrix}\end{aligned}\]
More practice problems: change of variables in multiple integrals ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-001 For the change x=2u,\ y=3v, what is |J|=|(∂(x,y))/(∂(u,v))|?ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-002 For x=u+v,\ y=u-v, the value of |J| is:ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-003 For polar coordinates x=rcos θ,\ y=rsin θ, the Jacobian (∂(x,y))/(∂(r,θ)) is:ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-004 If x=u^2-v^2,\ y=2uv, the Jacobian (∂(x,y))/(∂(u,v)) is:ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-005 With x=2u,\ y=3v, the area of the ellipse tfracx^24+tfracy^29≤1 is:ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-006 If u=x+y,\ v=x-y, then dx dy=ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-007 If at a point J=-3, what does the transformation do near it?ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-008 Let R be the region given by x=u+v,\ y=u-v with 0≤ u,v≤1. Compute iint_R (x^2-y^2) dA.