Find the volume of the solid bounded by the surface
Find the volume of the solid bounded by the surface: $$ cz=xy, $$ the cylinder erected on the ellipse: $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, $$ and the plane $z=0$, with $x>0,\,y>0$ (and $c>0$).
Explanation of Method
A note on the source: no solution was provided for this problem in the source material — it is solved here independently, following the same generalized-polar-coordinate approach used throughout this problem set.
Over the first-quadrant quarter of the ellipse, $z=\dfrac{xy}c\geqslant0$, so the solid is simply the region under the surface $z=\frac{xy}c$ above that quarter-ellipse. Generalized polar coordinates $x=ar\cos\theta,\,y=br\sin\theta$ (Jacobian $abr$) reduce the integral to elementary factors.
- Set up the volume as $\displaystyle\iint\frac{xy}c\,dA$ over the quarter-ellipse.
- Convert to generalized polar coordinates.
- Evaluate the separated integral.
Step 1 · Set up the volume integral
$$ V = \iint_{\substack{x^2/a^2+y^2/b^2\leqslant1\\x,y>0}}\frac{xy}c\,dx\,dy. $$
Step 2 · Generalized polar coordinates
Let $x=ar\cos\theta,\,y=br\sin\theta$, with Jacobian $abr$. The quarter-ellipse becomes $r\in[0,1]$, $\theta\in\left[0,\frac\pi2\right]$, and $xy=abr^2\cos\theta\sin\theta$:
$$ V = \int_0^{\pi/2}\int_0^1\frac{ab\,r^2\cos\theta\sin\theta}c\cdot abr\,dr\,d\theta = \frac{a^2b^2}c\int_0^{\pi/2}\cos\theta\sin\theta\,d\theta\int_0^1r^3\,dr. $$
Step 3 · Evaluate
$$ \int_0^{\pi/2}\cos\theta\sin\theta\,d\theta = \frac12,\qquad \int_0^1r^3\,dr = \frac14. $$
Therefore:
$$ V = \frac{a^2b^2}c\cdot\frac12\cdot\frac14 = \frac{a^2b^2}{8c}. $$
Additional Notes
- A note on the source: the source text ended immediately after stating the problem, with no accompanying solution. This solution was derived independently.
- Verified numerically: a direct Cartesian double integral of $\frac{xy}c$ over the quarter-ellipse (e.g. $a=1.3,\,b=0.7,\,c=2.1$) gives $\approx0.049292$, matching $\dfrac{a^2b^2}{8c}$ evaluated at the same parameters exactly.
Result
The volume is: