Hello mathematicians,
I cannot find the symmetry in the following expression (see image):
$$\int _{-3}^3(x^3 - 7x +2) \sqrt{9 - x^2}\, dx$$
How does the first component collapse (due to symmetry, I must suppose)? I do not see it. Any tips are much appreciated.
First, we note that $$\int _{-3}^3(x^3 - 7x +2) \sqrt{9 - x^2}\, dx = \int _{-3}^3 (x^3-7x)\sqrt{9-x^2}\,dx + \int_{-3}^{3}2\sqrt{9-x^2}\,dx.$$ Furthermore, if we consider $$f(x) = (x^3-7x)\sqrt{9-x^2}$$ then we can observe that \begin{align*} f(-x) &= ((-x)^3-7(-x))\sqrt{9-(-x)^2}\\&= (-x^3+7x)\sqrt{9-x^2}\\&= -(x^3-7x)\sqrt{9-x^2}\\&= -f(x) \end{align*} Since then we have $f(-x) = -f(x)$, we can conclude that $\int_{-3}^{3}f(x)\,dx = 0$, and so we have that $$\int _{-3}^3(x^3 - 7x +2) \sqrt{9 - x^2}\, dx = \int_{-3}^{3}2\sqrt{9-x^2}\,dx.$$