Computing area with parametrically represented boundaries : If the boundary of a figure is represented by parametric equation, i.e., `x=x(t), 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.
If the curve given by parametric equation `x=t-t^(3), y=1-t^(4)` forms a loop for all values of `t in [-1,1]` then the area of the loop is

Options

A. `(1)/(7)` sq. units
B. `(3)/(5)` sq. units
C. `(16)/(35)` sq. units
D. `(8)/(35)` sq. units

Answer

C

Explanation

`(dx)/(dt)=1-3t^(2)and (dy)/(dt)=1-4t^(3)`
`"So, "x(dy)/(dt)-y(dx)/(dt)=(t-t^(3))(1-4t^(3))-(1-t^(4))(1-3t^(2))=t^(6)-3t^(4)-t^(3)+3t^(2)+t-1`
`therefore" Required area "=(1)/(2)overset(1)underset(-1)int(t^(6)-3t^(4)-t^(3)+3t^(2)+t-1)dt`
`=(16)/(35)` sq. units (taking absolute value).