Exercise · ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-010 · ★★★ Advanced
Find the area of the region \(1\le xy\le2,\ 1\le y/x\le2\) using \(u=xy,\ v=y/x\) (positive quadrant).
Choose one option.
Show answer
- A. \(\tfrac{\ln2}{2}\)
Explanation
From \(x=\sqrt{u/v},\ y=\sqrt{uv}\) we get \(|J|=\tfrac1{2v}\), and the area is \(\int_1^2\!\int_1^2\tfrac{1}{2v}\,dv\,du=\tfrac{\ln2}{2}\). The rectangle \([1,2]^2\) in \(uv\) would have area \(1\), but the factor \(\tfrac1{2v}\) corrects it.
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-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.