Computing double integral over the bounded region

Question

I have to compute $ \int \int_D dxdy$ over the region bounded by $x^2=ay$, $x^2=by$, $y^2=cx$, $y^2=dx$, where $0<a<b$ and $0<c<d$.

When I look at the problem from one-dimensional view and try to compute the integral as the area between 2 functions, clearly I get answer as sum/difference of $a$,$b$,$c$,$d$ in some rational powers. But I want to use another approach by converting the problem to another system of coordinates, to be more precise on $u=x^2$, $v=y^2$. The aim is to simplify the limits of integration, and I assume that I have the next equivalent definite integral: $\int_a^b \int_c^d \dfrac{du dv}{4uv}$. The question is that I get the answer in the terms of logarithm functions.

I suppose that the mistake lies in the wrong change of variables, but I'm not sure how to fix it.

Answer 1

Although there is already an answer, I would suggest you do try this with new variables. You just need to pick them more carefully and I'll try to show you how.

For two of the given curves, $x^2=ay$ and $x^2=by$, the expression $x^2/y$ is a constant. Let $u=x^2/y$ and $u$ will run from $a$ to $b$.

Similarly, for the curves corresponding to $y^2=cx$ and $y^2=dx$, the expression $y^2/x$ is a constant. Let $v=y^2/x$ and $v$ will run from $c$ to $d$.

With these new variables $u$ and $v$, you will have constant limits for both variables! And even the Jacobian turns out to be very simple ($1/3$, you can check); so the integral becomes:

$$\iint_D \; dxdy = \int_a^b \int_c^d \frac{1}{3} \; dvdu = \frac{(b-a)(d-c)}{3}$$ Once you get the idea, you put a bit of time in choosing handy new variables and computing the new limits and Jacobian, but the calculations can become a lot easier.

Answer 2

With those substitutions, you would have the bounding functions $u^2=a^2v$, $u^2=b^2v$, $v^2=c^2u$, and $v^2=d^2u$. You are just changing the constants, not simplifying any integrals.

Instead, I would handle this by breaking down the domain of integration into $3$ regions

where the red region is between $y^2=cx$ and $x^2=ay$ and $a^2c\le x^3\le a^2d$, the green region is between $y^2=cx$ and $y^2=dx$ and $a^2d\le x^3\le b^2c$, and the blue region is between $x^2=by$ and $y^2=dx$ and $b^2c\le x^3\le b^2d$.