Question

The area of the region bounded by the curve $y=x^2-3x$ with $y\le 0$ is

• A. $3$
• B. $\frac{9}{2}$
• C. $\frac{5}{2}$
• D. $noneofthese$

Answer 1

The correct option is B $\frac{9}{2}$

REF.Image

when x=0 $\Rightarrow$ y0

when y=0 $\Rightarrow$ x(x-3)=0

x=0 and x=3

$y=x^2-3x$

$\frac{dy}{dx}=2x-3\Rightarrow x=1.5$

min value at x=1.5

$y=\frac{3}{2}(\frac{3}{2}-3)$

$=\frac{3}{2}\left(\frac{-3}{2}\right)=\frac{-9}{4}=-2.25$

Area of shaded region = $\int y.dx$

$={\int }_0^3(x^2-3x)dx$

$=(\frac{x^3}{3}-\frac{3x^2}{2}{)}_0^3$

$=\frac{3^3}{3}-3\times \frac{3^2}{2}=9-\frac{9\times 3}{2}=-4.5$

considering magnitude = 4.5