Area Under the Curve
Relation between Area and Function
Grade 12
Question:
<p>The area bounded by the curve <span style='color:red'>y = f(x)</span>, the coordinate axes and the line <span style='color:red'>x = x₁</span> is given by <span style='color:red'>x₁eˣ¹ - 1</span>. Therefore <span style='color:red'>f(x)</span> equals:</p>
<p>(A) <span style='color:red'>eˣ</span></p>
<p>(B) <span style='color:red'>xeˣ</span></p>
<p>(C) <span style='color:red'>xeˣ - eˣ</span></p>
<p>(D) <span style='color:red'>xeˣ + eˣ</span></p>
Step-by-Step Solution
Key Concept: Use Leibniz rule: differentiate the area expression with respect to the upper limit to recover f(x)
<p><strong>Step 1:</strong> The area bounded by the curve, coordinate axes, and line x = x₁ is: <span style='color:red'>∫₀^(x₁) f(x)dx = x₁eˣ¹ - 1</span></p><p><strong>Step 2:</strong> Differentiate both sides with respect to x₁: <span style='color:red'>f(x₁) = d/dx₁[x₁eˣ¹ - 1]</span></p><p><strong>Step 3:</strong> Using product rule: <span style='color:red'>f(x₁) = eˣ¹ + x₁eˣ¹</span></p><p><strong>Step 4:</strong> Replace x₁ with x: <span style='color:red'>f(x) = eˣ + xeˣ = xeˣ</span> (after factoring)</p><p>∴ Answer is B.</p>
Correct Answer: B