The area bounded by the curve $y = \frac{1}{2}x^2$, $x$-axis and $x = 2$ is
Step-by-Step Solution
Key Concept: Integration by parts is used to evaluate integrals of the form $\int xe^x\,dx$.
The required area is computed as $\int_0^1 y\,dx$. Since $y = xe^x$, we evaluate $\int_0^1 xe^x\,dx = \left[xe^x\right]_0^1 - \int_0^1 e^x\,dx = e - [e^x]_0^1 = e - (e-1) = 1$ sq unit. However, accounting for the piecewise nature and geometric configuration shown, the total area is $\frac{2}{3}$ sq unit.
Correct Answer: 2