I want to evaluate the integral $$\int\int \frac{x}{\sqrt{x^2+y^2}}dA$$ on a region R where R is the region enclosed by the 2 circles $$x^2+y^2=a^2$$ $$x^2+y^2=b^2$$ $$a<b$$ My question is : Do we have to use the symmetry ? or not ? My teacher said that we MUST use the symmetry in all problems of multiple integrals if we have symmetry over the region of integration , otherwise it will be wrong! But I do not know why ?! I am not convinced ..
I used polar coordinates , if I do not use symmetry $$\int_{\theta=0}^{\theta=2\pi}\int_{r=a}^{r=b}\cos(\theta)\ r \ dr\ d\theta \ =0$$ If I use symmetry : $$4\int_{\theta=0}^{\theta=\pi/2}\int_{r=a}^{r=b}\cos(\theta)\ r \ dr\ d\theta \ =2(b^2-a^2)$$
Which solution is right and why ? I think that we CAN NOT use symmetry here because the function $f(x,y)$ inside the integration is not even ! also when we used the polar coordinates : we have $\cos(\theta)$ in the integration which has values that do not repeat themselves for each quadrant! I see that we can use symmetry if the function inside integration does not depend on theta only ..
I think that we generally do not have to use symmetry except if we have symmetry in the function also not only the region of integration as my teacher said! I think that symmetry can help us to get the solution quickly but if we use it in a right way ..