Answer:
4.5 sq. units.
Step-by-step explanation:
The given curve is [tex]y = (3x)^{\frac{1}{2} }[/tex]
⇒ [tex]y^{2} = 3x[/tex] ...... (1)
This curve passes through (0,0) point.
Now, the straight line is y = 3x - 6 ....... (2)
Now, solving (1) and (2) we get,
[tex]y^{2} - y - 6 = 0[/tex]
⇒ (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 = [tex]\int\limits^3_0 {(3x)^{\frac{1}{2} } } \, dx - \int\limits^3_2 {(3x - 6)} \, dx[/tex]
= [tex]\sqrt{3} [{\frac{x^{\frac{3}{2} } }{\frac{3}{2} } }]^{3} _{0} - [\frac{3x^{2} }{2} - 6x ]^{3} _{2}[/tex]
= [tex][\frac{\sqrt{3}\times 2 \times 3^{\frac{3}{2} } }{3}] - [13.5 - 18 - 6 + 12][/tex]
= 6 - 1.5
= 4.5 sq. units. (Answer)
