Answer
a)
$\int_α^β \sqrt (f'(t))^{2}+(g'(t))^{2})dt $
b)
$ \int_α^β 2πg(t) \sqrt (f'(t))^{2}+(g'(t))^{2})dt $
Work Step by Step
a)
The length of a parametric curve is defined as $L= \int_α^β \sqrt ((\frac{dx}{dt})^{2}
+(\frac{dy}{dt})^{2})dt = \int_α^β \sqrt (f'(t))^{2}+(g'(t))^{2})dt $
b)
The area of the surface obtained by rotating a parametric curve about the x-axis is: $S= \int_α^β 2πy\sqrt ((\frac{dx}{dt})^{2}+(\frac{dy}{dt})^{2})dt = \int_α^β 2πg(t) \sqrt (f'(t))^{2}+(g'(t))^{2})dt $
