<p>Find the area bounded by the curve satisfying <span>\(\frac{dy}{dx} = 2x + 1\)</span> that passes through <span>\((1, 2)\)</span>, the x-axis from <span>\(x = 0\)</span> to <span>\(x = 1\)</span>.</p>
Step-by-Step Solution
Key Concept: Integrate the given differential equation, apply the initial condition to find the constant, then calculate the area under the curve.
<p><strong>Step 1:</strong> Integrate the differential equation: <span>$y = x^2 + x + c$</span></p><p><strong>Step 2:</strong> Use the initial condition <span>$(1, 2)$</span>: <span>$2 = 1 + 1 + c \Rightarrow c = 0$</span></p><p><strong>Step 3:</strong> The curve is <span>$y = x^2 + x$</span></p><p><strong>Step 4:</strong> Calculate the area: <span>$\text{Area} = \int_0^1 (x^2 + x) \, dx = \left[\frac{x^3}{3} + \frac{x^2}{2}\right]_0^1 = \frac{1}{3} + \frac{1}{2} = \frac{5}{6}$</span> sq units</p><p>∴ Answer is <span>$\frac{5}{6}$</span> sq units.</p>
Correct Answer: \frac{5}{6}