Question

The area of the region bounded by curves $y=x\log x$ and $y=2x-2x^2$ is

• A. $\frac{1}{2}$ sq units
• B. $\frac{3}{12}$ sq units
• C. $\frac{7}{12}$sq units
• D. None of these

Answer 1

The correct option is C

$\frac{7}{12}$sq units

Explanation for the correct option:

Determining the area under the curve.
Given,

$$\begin{gathered} y=x\log x \\ y=2x-2x^2 \end{gathered}$$

Representation of curves, as shown:

Required area $=$ shaded region in the diagram between $0$ and $1$(area enclosed$=$upper curve $-$lower curve)

. $$\begin{gathered} ={\int }_0^1\left(\left(2x-2x^2\right)-x\log x\right)dx \\ ={\left[\left(x^2-\frac{2}{3}{x}^3\right)-\left(\frac{x^2}{2}\log x-\frac{x^2}{4}\right)\right]}_0^1 \\ ={\left[x^2-\frac{2}{3}{x}^3\right]}_0^1-{\left[\frac{x^2}{2}\log x-\frac{x^2}{4}\right]}_0^1 \\ =\left[\left(1-\frac{2}{3}\right)-0\right]-\left[\left(x-\frac{1}{4}\right)-0\right] \\ =\frac{1}{3}+\frac{1}{4} \\ =\frac{7}{12}\text{sq.units.} \end{gathered}$$
Hence, the correct answer is option (C)