Solve the Integral in the following set

Question

I have the Set $D=\{ (x,y) \in\mathbb{R}: |x|\le2,|x|\le y \le \sqrt{4-x^2}\}$ and I have to calculate $\int_{D}(x^2y+xy^2) dydx$.

So I cannot explain how hard it is for me to find the borders of the integral. I've tried to understand it but It's very hard for meand every time I think i have understood it, i do it wrong in another excercise.

I have an exam tomorrow and of course I can't do wonders with understanding them. We often do them in the lectures that we turn them somehow to polar coordinates or so.

However, my question is: is there any way, that I can read it from the definition of the set D for example, with x and y(without having to turn them in some other coordinates) and what suggestions could you give me in order to be able to do that?

For example in this specific excercise, i thought that since $|x|\le2$ then $-2\le x\le 2 $. Then I just took $y$ between $2$ (since the absolute value of $x$ can't be more than $2$, and $\sqrt{4-x^2}$. So I calculated $\int_{-2}^{2}\int_{2}^{\sqrt{4-x^2}}x^2y$. My answer is not the same as the answer in the book, so I got completely lost.

I would be very thankful for some help.

Annalisa

Answer 1

$|x| \leq 2 \implies -2 \leq x \leq 2$

$|x| \leq y \leq \sqrt{4-x^2}$

$\implies -x \leq y \leq \sqrt{4-x^2} \ $ for $x \leq 0$ and $x \leq y \leq \sqrt{4-x^2} \ $ for $x \geq 0$.

So here is the sketch of the region (you are asked to integrate over the shaded region).

Please note that integral of $xy^2$ will cancel out over region for $x \geq 0$ and for $x \leq 0$ due to $x-$ symmetry.

Your integral is split into two as below -

$\displaystyle \int_{-\sqrt2}^{0} \int_{-x}^{\sqrt{4-x^2}} (x y^2 + yx^2) \ dy \ dx + \int_0^{\sqrt2} \int_{x}^{\sqrt{4-x^2}} (x y^2 + y x^2) \ dy \ dx$

As mentioned you can avoid integrating $xy^2$ as it will cancel out and will not change the result of your final integral.

Answer 2

If you read more carefully the definition of the domain $D$, then you will find that the given integral can be written as $$\iint_D(x^2y+xy^2)\,dxdy=\int_{x\in [-2,2], |x|\leq \sqrt{4-x^2}}\left(\int_{|x|}^{\sqrt{4-x^2}}(x^2y+xy^2)\,dy\right) \,dx$$ Next step: draw the graphs of the functions $|x|$ and $\sqrt{4-x^2}$ over the interval $[-2,2]$ and find explicitly the upper limit and the lower limit of the domain of integration with respect to $x$. Can you take it from here?

P.S. Now a big part of YOUR work (sketch the graphs + symmetry) has been done by Math Lover. This means that the interesting part of your exercise is gone... Well, that's a pity!