Question

The area of the region bounded by the parabola ( y^2 = 27x ) and the line ( x = 1 ) is ________ sq.units.

• A. ( 2\sqrt{3} )
• B. ( 3\sqrt{3} )
• C. ( 4\sqrt{3} )
• D. [suspicious link removed]

Hint:

Area of parabola $y^2 = 4ax$ from $x=0$ to $x=h$ is $\frac{8}{3}\sqrt{a}h^{3/2}$.

Answer 1

The Correct Option is C

Step 1: Setup the integral
Area is symmetric about the X-axis. Area (A = 2 \int_{0}^{1} y , dx).

Step 2: Substitute (y)
(y = \sqrt{27x} = 3\sqrt{3} \cdot x^{1/2}).

Step 3: Integration
(A = 2 \int_{0}^{1} 3\sqrt{3} x^{1/2} , dx = 6\sqrt{3} [ \frac{x^{3/2}}{3/2} ]_{0}^{1}).
(A = 6\sqrt{3} \cdot \frac{2}{3} [1 - 0] = 4\sqrt{3}).
Final Answer: (C)