Find the volume of the solid bounded by the surfaces
Find the volume of the solid bounded by the surfaces: $$ z=xy,\qquad xy=1,\qquad xy=4,\qquad y^2=x,\qquad y^2=3x,\qquad z=0. $$
Explanation of Method
The substitution $u=xy,\,v=y^2/x$ turns the four bounding curves into a coordinate rectangle $[1,4]\times[1,3]$, and the height $z=xy=u$ becomes trivial in the new coordinates, with Jacobian $|J|=\dfrac1{3v}$.
- Substitute $u=xy,\,v=y^2/x$, solve for $x,y$, and find the Jacobian.
- Express $z$ in the new coordinates.
- Set up and evaluate the volume integral over the rectangle.
Step 1 · Substitution and Jacobian
With $u=xy,\,v=y^2/x$: solving gives $x=u^{2/3}v^{-1/3},\,y=u^{1/3}v^{1/3}$, and the Jacobian works out to $\left|\dfrac{\partial(x,y)}{\partial(u,v)}\right|=\dfrac1{3v}$. The region becomes $u\in[1,4],\,v\in[1,3]$.
Step 2 · Express $z$
$z=xy=u$.
Step 3 · Set up and evaluate
$$ V=\int_1^3\int_1^4u\cdot\frac1{3v}\,du\,dv = \frac13\left[\int_1^4u\,du\right]\left[\int_1^3\frac{dv}v\right] = \frac13\cdot\frac{15}2\cdot\ln3 = \frac{5\ln3}2. $$
Additional Notes
- This derivation matches the source exactly.
- Verified numerically: direct evaluation of the $(u,v)$ double integral gives exactly $\dfrac{5\ln3}2$.
Result
The volume is: