How to find the surface area of a circle using definite integral in polar coordinates?

Question

I should find the surface area of this figure: $$r = \cos(\phi) - \sin(\phi)$$ I moved it to Cartesian coordinates and found out that it is the circle. When calculating it is necessary to use definite integral and polar coordinates. The integral must be over $d\phi$

Answer 1

The limits for $\phi$ are $0$ and $\pi$. The area is given by $$\int_0^{\pi} \frac{1}{2}r^2\,d\phi = \frac{1}{2}\int_0^{\pi}(\cos\phi-\sin\phi)^2\,d\phi = \frac{1}{2}\int_0^{\pi}(1-2\sin\phi\cos\phi)\,d\phi=\frac{1}{2}\pi.$$