Question

The region bounded by y=(3x)^(1/2), y=3x-6, y=0

Answer 1

4.5 sq. units.

Explanation:

The given curve is
`y = (3x)^{(1)/(2) }`

⇒
`y^(2) = 3x` ...... (1)

This curve passes through (0,0) point.

Now, the straight line is y = 3x - 6 ....... (2)

Now, solving (1) and (2) we get,

`y^(2) - y - 6 = 0`

⇒ (y - 3)(y + 2) = 0

⇒ y = 3 or y = -2

We will consider y = 3.

Now, y = 3x - 6 has zero at x = 2.

Therefor, the required are =
`\int\limits^3_0 {(3x)^{(1)/(2) } } \, dx - \int\limits^3_2 {(3x - 6)} \, dx`

=
`√(3) [{\frac{x^{(3)/(2) } }{(3)/(2) } }]^(3) _(0) - [(3x^(2) )/(2) - 6x ]^(3) _(2)`

=
`[\frac{√(3)* 2 * 3^{(3)/(2) }  }{3}] - [13.5 - 18 - 6 + 12]`

= 6 - 1.5

= 4.5 sq. units. (Answer)