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Computing area with parametrically repre...

Computing area with parametrically represented boundaries
If the boundary of a figure is represented by parametric equations x = x (t) , y = y(t) , then the area of the figure is evaluated by one of the three formulae
`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 traversal of the contour .
The area enclosed by the astroid `((x)/(a))^((2)/(3)) + ((y)/(a))^((2)/(3))` = 1 is

A

`(a^(2) pi)/(8)`

B

`(a^(2) pi)/(4)`

C

`(3pi a^(2))/(8)`

D

`(pi a^(2))/(2)`

Text Solution

Verified by Experts

The correct Answer is:
C
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