Question

Find the total area of the bounded region enclosed between the parabola curve \( y^2 = 8x \) and its vertical latus rectum boundary line.

• A. \( \frac{32}{3} \text{ square units} \)
• B. \( \frac{16}{3} \text{ square units} \)
• C. \( 8 \text{ square units} \)
• D. \( \frac{64}{3} \text{ square units} \)

Hint:

You can save valuable time during your test by using this parabola integration shortcut: the total area enclosed between any standard parabola \( y^2 = 4ax \) and its latus rectum line is always exactly \( \frac{8}{3}a^2 \).

Answer 1

The Correct Option is A

Concept: The standard parabola equation \( y^2 = 4ax \) features a focal point at coordinates \( (a,0) \). Its latus rectum is the vertical boundary line running perpendicular to its axis through this focus, defined by the line equation \( x = a \).

Step 1: Identify the boundary limit coordinates.
Compare our given equation \( y^2 = 8x \) to the standard parabola layout \( y^2 = 4ax \): \[ 4a = 8 \quad \Rightarrow \quad a = 2 \] This places our vertical latus rectum boundary line at \( x = 2 \). The integration limits for the bounded region run from the origin vertex (\( x = 0 \)) to this boundary line (\( x = 2 \)).

Step 2: Set up the integration equation using symmetry properties.
Because the parabola curve is perfectly symmetric across the x-axis, calculate the total area by doubling the area of the upper quadrant slice: \[ \text{Area} = 2 \int_{0}^{2} y \, dx = 2 \int_{0}^{2} \sqrt{8x} \, dx = 2 \cdot \sqrt{8} \int_{0}^{2} x^{\frac{1}{2}} \, dx = 4\sqrt{2} \int_{0}^{2} x^{\frac{1}{2}} \, dx \]

Step 3: Evaluate the definite integral using the power rule.
\[ \text{Area} = 4\sqrt{2} \left[ \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right]_{0}^{2} = 4\sqrt{2} \cdot \frac{2}{3} \left[ x^{\frac{3}{2}} \right]_{0}^{2} = \frac{8\sqrt{2}}{3} \left( 2^{\frac{3}{2}} - 0 \right) \] Since \( 2^{\frac{3}{2}} = 2\sqrt{2} \), substitute this back into the expression: \[ \text{Area} = \frac{8\sqrt{2}}{3} \cdot 2\sqrt{2} = \frac{16 \cdot 2}{3} = \frac{32}{3} \text{ square units} \]