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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 = -underset(alpha)overset(beta)(int) y(t) x'(t) dt , S = underset(alpha) overset(beta) (int) x (t) y' (t) dt`
`S = (1)/(2) underset(alpha)overset(beta)(int) (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)`3/4 a^2 pi`

B

(b)`3/18 pi a^(2)`

C

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

D

(d)`3/4 a pi`

Text Solution

Verified by Experts

The correct Answer is:
C
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