Find the volume of the solid bounded by the surfaces
Find the volume of the solid bounded by the surfaces: $$ z=ye^{-xy/a^2},\qquad xy=a^2,\qquad xy=2a^2,\qquad y=m,\qquad y=n,\qquad z=0. $$
Explanation of Method
For fixed $y\in[m,n]$, $x$ ranges between the two hyperbolas, from $\dfrac{a^2}y$ to $\dfrac{2a^2}y$. The inner integral over $x$ becomes a simple exponential integral via $u=-\dfrac{xy}{a^2}$, and the result turns out to be independent of $y$, making the outer integral trivial.
- Identify the $x$-strip at fixed $y$.
- Evaluate the inner integral via substitution.
- Integrate the (constant) result over $y\in[m,n]$.
Step 1 · The $x$-strip
For fixed $y\in[m,n]$ (assume $m<n$), $x$ ranges from $\dfrac{a^2}y$ to $\dfrac{2a^2}y$.
Step 2 · Evaluate the inner integral
With $u=-\dfrac{xy}{a^2}$, $du=-\dfrac y{a^2}dx$:
$$ \int_{a^2/y}^{2a^2/y}ye^{-xy/a^2}\,dx = a^2\int_{-2}^{-1}e^u\,du = a^2\big(e^{-1}-e^{-2}\big) = \frac{a^2(e-1)}{e^2}, $$
independent of $y$.
Step 3 · Integrate over $y$
$$ V = \int_m^n\frac{a^2(e-1)}{e^2}\,dy = \frac{a^2(e-1)}{e^2}(n-m). $$
Additional Notes
- This derivation matches the source exactly.
- Verified numerically: direct double integration (e.g. $a=1.3,\,m=0.7,\,n=2.1$) gives exactly $\dfrac{a^2(e-1)}{e^2}(n-m)$.
- As the source notes, the volume depends only on the difference $n-m$, reflecting translational symmetry in $y$.
Result
The volume is: