Exercise · ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-008 · ★★☆ Intermediate
Let \(R\) be the region given by \(x=u+v,\ y=u-v\) with \(0\le u,v\le1\). Compute \(\iint_R (x^2-y^2)\,dA\).
Choose one option.
Show answer
- C. \(2\)
Explanation
\(x^2-y^2=4uv\) and \(|J|=2\), so the integral is \(\int_0^1\!\int_0^1 8uv\,du\,dv=2\). Forgetting \(|J|\) gives \(1\).
Review the concept: Change of variables in multiple integrals
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-009 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)/(…
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