I’m converting integrals into polar integrals, but I don’t understand how to convert, for example, an integral that runs on the upper half of a circle from $-1$ to $2$ on the $x$-axis.
What’s confusing me is that all the examples I’ve been given have $x$ run from $0$ or $-r$ to $r$.
in polar coordinates:
$$x = rcos\theta$$ $$y = rsin\theta$$
we have that
$$x^2+y^2= r^2$$ $$r^2cos^2\theta+r^2sin^2\theta = r^2$$
$$dA = \frac{1}{2}r^2d\theta$$
$$A = \int_{\theta_0}^{\theta}\frac{1}{2}r^2d\theta$$ $$A = \int_{\theta_0}^{\theta}\frac{1}{2}(r^2cos^2\theta+r^2sin^2\theta )d\theta$$
because we are integrating tiny triangles with height $d\theta$ and base $r^2$