Find area bounded by the curve `sqrt(x) +sqrt(y) =sqrt(a)` & coordinate axes.

Options

A. `(5a^(2))/(6)`
B. `(a^(2))/(3)`
C. `(a^(2))/(2)`
D. `(a^(2))/(6)`

Answer

D

Explanation

Area bounded by curve `sqrt(x) + sqrt(y) = sqrt(a) (x,y ge 0)` and coordinate axes is
`= int_(0)^(a) y dx = int_(0)^(a) a + x - 2 sqrt(a) sqrt(x) dx`
`( :' sqrt(y) = sqrt(a) - sqrt(x) rArr y = a + x - 2 sqrt(a) sqrt(x))`
`= ax+(x^(2))/(2) - (2sqrt(a) x^(3/2))/(3/2) |_(0)^(a)`
`= a^(2) + (a^(2))/(2) - (4)/(3)a^(2) = (3a^(2))/(2) - (4)/(3) a^(2)`
`= (9a^(2) - 8a^(2))/(6) = (a^(2))/(6)` sq. unit