Differential Equations
Linear ODE — area enclosed
Grade Class 12
Question:
<p>Area enclosed between \\(y=xe^{-x}\\) and \\(x\\)-axis for \\(x\\in[0,\\infty)\\). Find area.</p>
Step-by-Step Solution
Key Concept: Integer/Numeric entry — compute exact numerical answer.
<div class='solution'><p>$\int_0^\infty xe^{-x}\,dx$. Integration by parts: $= [-xe^{-x}]_0^\infty + \int_0^\infty e^{-x}\,dx = 0 + [-e^{-x}]_0^\infty = 0-(-1)=1$. Answer: <strong>1</strong>.</p></div>
Correct Answer: 1