# The curve with equation y=x(3-x)^(1/2) together with line segment OA.!) what are the coordinates of B and A; 2)what is the Area of the shaded region bounded by the line segment AO, x axis and the arc of AB curve?

Jul 28, 2018

The coordinates of $B = \left(2 , 0\right)$. The point $A = \left(2 , 2\right)$. The area is

$= 3.6 {u}^{2}$

#### Explanation:

Calculation of the point $B$

Let

$y = x \sqrt{3 - x} = 0$

Then,

$\left\{\begin{matrix}x = 0 \text{ this is the origin "O \\ x=3 " this is the point B}\end{matrix}\right.$

The coordinates of $B = \left(2 , 0\right)$

Calculation of the point $A$

Calculate the derivative of $y$ according to the product rule

$\frac{\mathrm{dy}}{\mathrm{dx}} = {\left(3 - x\right)}^{\frac{1}{2}} - \frac{x}{2 {\left(3 - x\right)}^{\frac{1}{2}}}$

$= \frac{2 \left(3 - x\right) - x}{3 - x} ^ \left(\frac{1}{2}\right)$

$= \frac{6 - 3 x}{3 - x} ^ \left(\frac{1}{2}\right)$

The maximum is when

$\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

That is

$6 - 3 x = 0$, $\implies$, $x = 2$

The point $A = \left(2 , 2\right)$

The equation of the line $O A$ is

$y - 0 = 1 \left(x - 0\right)$, $\implies$, $y = x$

The area of the shaded region is

$A = {\int}_{0}^{2} x \mathrm{dx} + {\int}_{2}^{3} x \sqrt{3 - x} \mathrm{dx}$

$= {\int}_{2}^{3} x \sqrt{3 - x} + {\int}_{0}^{2} x \mathrm{dx}$

$= {I}_{1} + {I}_{2}$

Calculation of ${I}_{1}$ by substitution

Let $u = 3 - x$, $\implies$, $\mathrm{du} = - \mathrm{dx}$

$x = 3 - u$

Therefore,

${I}_{1} = {\int}_{2}^{3} x \sqrt{3 - x} \mathrm{dx} = {\int}_{1}^{0} \left(3 - u\right) \sqrt{u} - \mathrm{du}$

$= {\int}_{1}^{0} {u}^{\frac{3}{2}} - 3 {u}^{\frac{1}{2}} \mathrm{du}$

$= {\left[\frac{2}{5} {u}^{\frac{5}{2}} - 2 {u}^{\frac{3}{2}}\right]}_{1}^{0}$

$= \left(0\right) - \left(\frac{2}{5} \cdot {1}^{\frac{5}{2}} - 2 \cdot {1}^{\frac{3}{2}}\right)$

$= \frac{8}{5}$

$= 1.6$

${I}_{2} = {\int}_{0}^{2} x \mathrm{dx} = {\left[{x}^{2} / 2\right]}_{0}^{2}$

$= 2$

Therefore, the area is

$A = 1.6 + 2 = 3.6 {u}^{2}$

graph{(y-xsqrt(3-x))(y-x)=0 [-0.96, 5.197, -0.3, 2.778]}