Exercise · ZW-CHANGE-OF-VARIABLES-IN-MULTIPLE-INTEGRALS-001 · ★☆☆ Basic
For the change \(x=2u,\ y=3v\), what is \(|J|=\left|\frac{\partial(x,y)}{\partial(u,v)}\right|\)?
Choose one option.
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- B. \(6\)
Explanation
\(J=\begin{vmatrix}2&0\\0&3\end{vmatrix}=6\): each rectangle of the \(uv\)-plane is stretched six times in area.
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
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