(Solved): Calculate the area of the region of the Cartesian plane bounded by the indicated curves. Verify that ...

(2) Calculate the area of the region of the Cartesian plane bounded by the indicated curves. Verify that the intersection points are as indicated (in some cases, simple algebra is required). (a) \( f(x)=\sqrt{x}, g(x)=2 x \). (Hint: the region is delimited by intersection points \( P\left(\frac{1}{4}, \frac{1}{2}\right) \) and the origin) (b) \( f(x)=-2 x^{2}-x+1, g(x)=x^{2}+x+1 \). (Hint: the region is delimited by the two intersection points \( A\left(-\frac{2}{3}, \frac{7}{9}\right) \) and \( \left.B(0,1)\right) \) \( \left(\mathrm{c}^{*}\right) f(x)=\sin (2 x), g(x)=\sin x \), in the interval \( X=[-\pi / 2, \pi / 2] \). (Hint: There are two intersection points delimiting the region in \( X \) : \( A\left(-\frac{\pi}{3},-\frac{\sqrt{3}}{2}\right) \) and \( B\left(\frac{\pi}{3}, \frac{\sqrt{3}}{2}\right) \) Note that the area is symmetric with respect to rotation by \( \pi \) about the origin.)