Question

The area bounded by the parabola $y^2 = 4ax$ and its latus-rectum $x = a$ is

• A. $\frac{8}{3} a^2$ sq. units
• B. $\frac{2}{3} a^2$ sq. units
• C. $\frac{4}{3} a^2$ sq. units
• D. $8 a^2$ sq. units

Hint:

This is a classic standard area tracking formula in integral calculus! The area bounded by a standard parabola $y^2 = 4ax$ and its vertical latus-rectum line $x=a$ is always globally fixed as exactly $\frac{8}{3}a^2$. Memorizing this specific standard result can save you over a minute of manual integration during exams.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The problem asks for the total geometric surface area enclosed between the standard horizontal parabola $y^2 = 4ax$ and its focal vertical boundary line (the latus-rectum) given by $x = a$.

Step 2: Key Formula or Approach:
The parabola is perfectly symmetrical across the horizontal $x$-axis. Therefore, the total area is equal to two times the definite integral calculated for the upper half (from $y = 0$ to $y = \text{top curve}$) across the horizontal limits from $0$ to $a$: $$\text{Area} = 2 \int_{0}^{a} y \, dx = 2 \int_{0}^{a} \sqrt{4ax} \, dx$$

Step 3: Detailed Explanation:
1. Express $y$ explicitly in terms of $x$ for the upper curve quadrant: $$y^2 = 4ax \implies y = 2\sqrt{a}\cdot\sqrt{x} = 2\sqrt{a}\cdot x^{1/2}$$ 2. Set up the definite integral for total area taking symmetry into account: $$\text{Area} = 2 \int_{0}^{a} 2\sqrt{a}\cdot x^{1/2} dx$$ $$A = 4\sqrt{a} \int_{0}^{a} x^{1/2} dx$$ 3. Integrate using the standard power integration rule ($\int x^n dx = \frac{x^{n+1}}{n+1}$): $$A = 4\sqrt{a} \left[ \frac{x^{3/2}}{\frac{3}{2}} \right]_{0}^{a} = 4\sqrt{a} \cdot \left(\frac{2}{3}\right) \left[ x^{3/2} \right]_{0}^{a}$$ $$A = \frac{8}{3}\sqrt{a} \cdot \left( a^{3/2} - 0 \right)$$ 4. Combine the variable terms using index laws ($\sqrt{a} \cdot a^{3/2} = a^{1/2} \cdot a^{3/2} = a^2$): $$A = \frac{8}{3} a^2\text{ sq. units}$$

Step 4: Final Answer:
The bounded area is $\frac{8}{3} a^2$ sq. units, which matches option (A).