Question

What is the area bounded by the curve $y=\frac{1}{2}{x}^2$, the $X-axis$ and the line $x=2$ .

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $1$
• D. $\frac{4}{3}$

Answer 1

The correct option is D

$\frac{4}{3}$

Explanation of correct answer :

Finding area bounded by all curves:

The graph is as shown:

The shaded portion is the area bounded by $y=\frac{1}{2}{x}^2$, $X-axis$, $x=2$.

Required area $={\int }_0^{2}ydx$

$$\begin{gathered} ={\int }_0^2\frac{x^2}{2}dx \\ ={\left[\frac{x^3}{6}\right]}_0^2 \\ =\frac{8}{6} \\ =\frac{4}{3} \end{gathered}$$

Thus, the required area is $\frac{4}{3}$ sq units.

Hence, the correct answer is Option(A).