Using an appropriate change of variables, calculate

Using an appropriate change of variables, calculate the area bounded by the curves: $$ y^2=2px,\qquad y^2=2qx,\qquad x^2=2ry,\qquad x^2=2sy, $$ where $0<p<q$ and $0<r<s$.

Step 1 · Substitute and identify the region

Let $u=\dfrac{y^2}{2x},\,v=\dfrac{x^2}{2y}$. Then $y^2=2px\Rightarrow u=p$, $y^2=2qx\Rightarrow u=q$, $x^2=2ry\Rightarrow v=r$, and $x^2=2sy\Rightarrow v=s$. So the region maps to the rectangle $[p,q]\times[r,s]$ in the $(u,v)$ plane.

Step 2 · Invert and compute the Jacobian

From $y^2=2ux$ and $x^2=2vy$: substituting $y=\dfrac{x^2}{2v}$ into $y^2=2ux$ gives $\dfrac{x^4}{4v^2}=2ux\Rightarrow x^3=8uv^2\Rightarrow x=2u^{1/3}v^{2/3}$, and then $y=\dfrac{x^2}{2v}=2u^{2/3}v^{1/3}$. Differentiating:

$$ \frac{\partial x}{\partial u}=\frac{2v^{2/3}}{3u^{2/3}},\quad \frac{\partial x}{\partial v}=\frac{4u^{1/3}}{3v^{1/3}},\quad \frac{\partial y}{\partial u}=\frac{4v^{1/3}}{3u^{1/3}},\quad \frac{\partial y}{\partial v}=\frac{2u^{2/3}}{3v^{2/3}}. $$

$$ \frac{\partial(x,y)}{\partial(u,v)} = \frac{2v^{2/3}}{3u^{2/3}}\cdot\frac{2u^{2/3}}{3v^{2/3}} - \frac{4u^{1/3}}{3v^{1/3}}\cdot\frac{4v^{1/3}}{3u^{1/3}} = \frac49-\frac{16}9 = -\frac43, $$

so $\left|\dfrac{\partial(x,y)}{\partial(u,v)}\right|=\dfrac43$ (a constant), and $dx\,dy=\dfrac43\,du\,dv$.

Step 3 · Evaluate the area integral

$$ A = \int_p^q\!\int_r^s\frac43\,dv\,du = \frac43(s-r)(q-p). $$

Additional Notes

• This derivation matches the source exactly. The elegant result — a constant Jacobian $\frac43$ — means the curvilinear quadrilateral's area is simply $\frac43$ times the product of the two parameter differences, regardless of $p,q,r,s$ individually.
• Verified numerically for $p=0.5,\,q=1.4,\,r=0.3,\,s=1.1$: the formula gives $\frac43(q-p)(s-r)=\frac43\times0.9\times0.8=0.96$. This was cross-checked independently by tracing the actual curvilinear-quadrilateral boundary in the $(x,y)$ plane (via the inverse map $x=2u^{1/3}v^{2/3},\,y=2u^{2/3}v^{1/3}$ applied to the rectangle's edges) and applying the shoelace formula, which gives $A\approx0.960000$ — matching to 6 significant figures.