Applications of definite integral

Exercise

1.: Find the area of the figure surrounded by the following curves．
(a): $\displaystyle{x = y^{2} , x = 3 -2y^{2}}$
(b): $\displaystyle{x = \cos^{3}{t}, y = \sin^{3}{t}, \ (0 \leq t \leq \pi)}$ and -axis
2.: Find the volume of a rotating body formed by rotating the following plane figure around the axis
(a): $\displaystyle{x^{2}+ (y-2)^{2} \leq 1}$
(b): $\displaystyle{\ x = t - \sin{t}, y = 1 - \cos{t}, \ 0 \leq t \leq 2\pi}$ and -axis．
3.: Find the length of the following curves．
(a): $\displaystyle{x^{2/3} + y^{2/3} = 1}$
(b): $\displaystyle{\sqrt{x} + \sqrt{y} = 1}$
(c): $\displaystyle{r = 1+\cos{\theta}}$