Computing area with parametrically represented boundaries : If the boundary of a figure is represented by parametric equation, i.e., `x=x(t), y=y(t),` then the area of the figure is evaluated by one of the three formulas :
`S=-int_(alpha)^(beta)y(t)x'(t)dt,`
`S=int_(alpha)^(beta)x(t)y'(t)dt,`
`S=(1)/(2)int_(alpha)^(beta)(xy'-yx')dt,`
Where `alpha and beta` are the values of the parameter t corresponding respectively to the beginning and the end of the traversal of the curve corresponding to increasing t.
The area of the region bounded by an are of the cycloid `x=a(t-sin t), y=a(1- cos t)` and the x-axis is

Options

A. (a)`6pia^(2)` sq. units
B. (b)`3pia^(2)` sq. units
C. (c)`4pia^(2)` sq. units
D. (d)None of these

Answer

B

Explanation

`S=|-overset(2pi)underset(0)inta(1-cos t)a(1- cos t) dt|`
`=|-a^(2)overset(2pi)underset(0)int(1-2 cos t + cos^(2)t ) dt|`
`=|-a^(2)overset(2pi)underset(0)int(1-2 cos t+((1+cos 2 t)/(2)))dt|`
`=|-(a^(2))/(2)overset(2pi)underset(0)int(3-4cos t+cos 2t)dt|`
`=|-(a^(2))/(2)[3t-4cos t +cos 2t]_(0)^(2pi)|`
`=|-3pia^(2)|=3pia^(2)` sq. units.