Question

What is the area bounded by the curve $y=2x-x^2$ and the line $y=-x$.

• A. $\frac{3}{2}$sq units
• B. $\frac{9}{3}$sq units
• C. $\frac{9}{2}$sq units
• D. None of these

Answer 1

The correct option is C

$\frac{9}{2}$sq units

Explanation of correct answer :

Finding the area within the curves:

The graph of the curves is shown as follows:

The shaded portion is the area within the curves.

The equations of given curve and a line are $y=2x-x^2\dots \left(1\right)$ and $y=-x\cdots \left(2\right)$ respectively. Solve both equations.

$$\begin{gathered} -x=2x-x^2 \\ \Rightarrow x^2-3x=0 \\ \Rightarrow x(x-3)=0 \\ \Rightarrow x=0\text{and}x=3 \end{gathered}$$

Putting the values of $x$ in $\left(2\right)$, we get two points that are $(0,0)\text{and}(3,3)$.

The point of intersection after solving both the equations are $(0,0)\text{and}(3,3)$.

The required area $={\int }_0^3\left[\left(2x-x^2\right)-\left(-x\right)\right]dx$

$$\begin{gathered} ={\int }_0^3\left(3x-x^2\right)dx \\ =\left[\frac{3x^2}{2}-\frac{x^3}{3}\right]dx \\ =\frac{27}{2}-\frac{27}{3} \\ =\frac{9}{2} \end{gathered}$$

The, area within the curve is $\frac{9}{2}$sq units.

Hence, the correct answer is Option(C).