Question

Find the surface area limited by √x + √y = √a and the coordinate axes

Answer 1

`(a^(2) )/(6)` square units.

Explanation:

The given curve is √x + √y = √a ......... (1)

Now, the curve intersects the x-axis at (a,0) point and the y-axis at (0,a) point.

Therefore, the are limited by equation (1) and the coordinate axes will be

=
`\int\limits^a_0 {(√(a) -√(x)  )^(2) } \, dx`

=
`\int\limits^a_0 {a+x+2√(ax)  } \, dx`

=
`[ax + (x^(2) )/(2)+ 2√(a)\frac{x^{(3)/(2) } }{(3)/(2) }  ]_(0) ^(a)`

=
`a^(2)+(a^(2) )/(2)- (4a^(2) )/(3)`

=
`(a^(2) )/(6)` square units. (Answer)