Answer
(a) Length of a parametric curve can be 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) Area of the surface is obtained by rotating a parametric curve about the $x$-axis .That is: $$S= \int_α^β 2πy\sqrt ((\frac{dx}{dt})^{2}+(\frac{dy}{dt})^{2})dt = \int_α^β 2πg(t) \sqrt (f'(t))^{2}+(g'(t))^{2})dt $$
Work Step by Step
(a) Length of a parametric curve can be 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) Area of the surface is obtained by rotating a parametric curve about the x-axis .That is: $$S= \int_α^β 2πy\sqrt ((\frac{dx}{dt})^{2}+(\frac{dy}{dt})^{2})dt = \int_α^β 2πg(t) \sqrt (f'(t))^{2}+(g'(t))^{2})dt $$
