Area Under the Curve
Area bounded by curves from differential condition
Grade 12
Question:
<p>The length of sub-normal at any point P(x, y) on the curve, which is passing through M(0, 1) is unity. The area bounded by the curves satisfying this condition is equal to</p>
<p>(A) \(\frac{1}{3}\)</p>
<p>(B) \(\frac{2}{3}\)</p>
<p>(C) \(\frac{4}{3}\)</p>
<p>(D) \(\frac{8}{3}\)</p>
Step-by-Step Solution
Key Concept: Sub-normal length $y\frac{dy}{dx} = 1$ defines a specific family of curves; integrate to find the area.
<p>The length of sub-normal is given by $y\frac{dy}{dx}$. Given that this equals unity and the curve passes through M(0, 1), we need to find the curve and then calculate the bounded area.</p>
Correct Answer: B