Area Under the Curve
Area division
Grade 12
Question:
<p>If the abscissa \(x = a\) divides the area bounded by the X-axis part of the curve \(y = 1 + \frac{8}{x^2}\) and the abscissa \(x = 2, x = 4\) into two equal parts, then \(a\) is equal to</p>
<p>(a) 2 sq units</p>
<p>(b) 2√2 sq units</p>
<p>(c) 3√2 sq units</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: To find the value of 'a' that divides the total area into two equal parts, we must calculate the total area under the curve, then find 'a' such that the area from x=2 to x=a equals half the total area.
<p><strong>Step 1: Find the total area under the curve from x=2 to x=4</strong></p><p>The area is: $$A_{total} = \int_2^4 \left(1 + \frac{8}{x^2}\right) dx$$</p><p><strong>Step 2: Evaluate the integral</strong></p><p>$$A_{total} = \left[x - \frac{8}{x}\right]_2^4$$</p><p>$$= \left(4 - \frac{8}{4}\right) - \left(2 - \frac{8}{2}\right)$$</p><p>$$= (4 - 2) - (2 - 4)$$</p><p>$$= 2 - (-2) = 4 \text{ sq units}$$</p><p><strong>Step 3: Set up the equation for dividing area into two equal parts</strong></p><p>The area from x=2 to x=a should equal half the total area:</p><p>$$\int_2^a \left(1 + \frac{8}{x^2}\right) dx = \frac{4}{2} = 2$$</p><p><strong>Step 4: Evaluate the integral from 2 to a</strong></p><p>$$\left[x - \frac{8}{x}\right]_2^a = 2$$</p><p>$$\left(a - \frac{8}{a}\right) - \left(2 - \frac{8}{2}\right) = 2$$</p><p>$$a - \frac{8}{a} - 2 + 4 = 2$$</p><p>$$a - \frac{8}{a} + 2 = 2$$</p><p>$$a - \frac{8}{a} = 0$$</p><p><strong>Step 5: Solve for a</strong></p><p>$$a = \frac{8}{a}$$</p><p>$$a^2 = 8$$</p><p>$$a = \sqrt{8} = 2\sqrt{2}$$</p><p>Since a must be in the interval (2, 4) and $2\sqrt{2} \approx 2.83$, this is valid.</p><p><strong>∴ Answer: B</strong></p>
Correct Answer: B