By introducing new variables u and v in place of x and y
By introducing new variables $u$ and $v$ in place of $x$ and $y$, determine the limits of integration for: $$ \int_a^b dx \int_{\alpha x}^{\beta x} z(x,y)\,dy \qquad (0<a<b;\ 0<\alpha<\beta), $$ using the substitution $u=x$, $v=\dfrac{y}{x}$.
Explanation of Method
The original region is a sector-like shape bounded by two lines through the origin ($y=\alpha x$ and $y=\beta x$) and two vertical lines ($x=a$, $x=b$). Since $v=y/x$ is exactly the slope of the ray from the origin through $(x,y)$, each of the two slanted boundaries becomes a constant value of $v$, turning the region into a plain rectangle.
- Transform each of the four boundaries into $(u,v)$-coordinates.
- Compute the Jacobian of the inverse transformation $x=u$, $y=uv$.
- Write the transformed integral over the resulting rectangle.
Step 1 · Transform the boundaries
With $u=x$, $v=\dfrac yx$ (so $x=u$, $y=uv$):
$$ x=a \Rightarrow u=a, \qquad x=b \Rightarrow u=b, $$
$$ y=\alpha x \Rightarrow v=\alpha, \qquad y=\beta x \Rightarrow v=\beta. $$
So the transformed region $D'$ is the rectangle $a\leqslant u\leqslant b$, $\alpha\leqslant v\leqslant\beta$.
Step 2 · Compute the Jacobian
With $x=u$, $y=uv$:
$$ \frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix}1&0\\v&u\end{vmatrix} = u, \qquad dx\,dy = u\,du\,dv. $$
Step 3 · Write the transformed integral
$$ \int_a^b dx\int_{\alpha x}^{\beta x} z(x,y)\,dy = \int_a^b\int_\alpha^\beta u\cdot z(u,uv)\,dv\,du. $$
Additional Notes
- This substitution converts a sector-like region bounded by rays from the origin into a rectangle with constant limits, which is the general strategy behind "polar-like" substitutions of the form $v=y/x$.
Result
The limits of integration and transformed integral are: