Applications of definite integral

Exercise

1.

Find the area of the figure surrounded by the following curves．

(a)

$${x = y^{2} , x = 3 -2y^{2}}$$

(b)

$${x = \cos^{3}{t}, y = \sin^{3}{t}, \ (0 \leq t \leq \pi)}$$ and $x$-axis

2.

Find the volume of a rotating body formed by rotating the following plane figure around the $x$ axis

(a)

$${x^{2}+ (y-2)^{2} \leq 1}$$

(b)

$${\ x = t - \sin{t}, y = 1 - \cos{t}, \ 0 \leq t \leq 2\pi}$$ and $x$-axis．

3.

Find the length of the following curves．

(a)

$${x^{2/3} + y^{2/3} = 1}$$

(b)

$${\sqrt{x} + \sqrt{y} = 1}$$

(c)

$${r = 1+\cos{\theta}}$$