Chapter 5, Problem 37RE

To determine

The value of the area under the curve.

Answer to Problem 37RE

The area under the curve is 12.8.

Explanation of Solution

Given information:

The equation of curve is $y=x\sqrt{x}$ (1)

The region lies between $x=0$ and $x=4$.

Calculation:

To find the value of y by using Equation (1) as shown below:

Substitute 0 for x in Equation (1).

$\begin{array}{c}y=0\sqrt{0}\\ =0\end{array}$

Substitute 1 for x in Equation (1).

$\begin{array}{c}y=1\sqrt{1}\\ =1\end{array}$

Similarly calculate the values of y for different x values as shown in Table (1).

x	y
0	0
1	1.00
2	2.83
3	5.20
4	8.00

Plot the curve using the values from Table (1).

Draw the diagram for the equation of curve as shown in Figure (1).

Refer Figure (1)

Here, area of the shaded portion 1 is $A_1$, area of the shaded portion 2 is $A_2$, area of the shaded portion 3 is $A_3$ and the area of the shaded portion 4 is $A_4$,

To find the area under the graph using Figure (1) as shown below:

$\begin{array}{c}\text{Area}=A_1+A_2+A_3+A_4\\ =\left(\frac{1}{2}\times 1\times 1\right)+\left(\frac{1}{2}\times 1\times \left(1+2.83\right)\right)+\left(\frac{1}{2}\times 1\times \left(2.83+5.2\right)\right)+\left(\frac{1}{2}\times 1\times \left(5.2+8\right)\right)\\ =0.5+1.915+4.015+6.6\\ =13.03\end{array}$

Therefore, the rough estimate of area under the graph is 13.03.

The expression to find the area under the curve as shown below:

$\begin{array}{c}{\displaystyle \underset{0}{\overset{4}{\int }}x\sqrt{x}dx}={\displaystyle \underset{0}{\overset{4}{\int }}{x}^{3/2}dx}\\ ={\left(\frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}\right)}_0^4\\ ={\left(\frac{x^{5/2}}{\frac{5}{2}}\right)}_0^4\\ =\frac{2}{5}\left(4^{5/2}-0\right)\end{array}$

$\begin{array}{c}{\displaystyle \underset{0}{\overset{4}{\int }}x\sqrt{x}dx}=\frac{2}{5}\times 32\\ =\frac{64}{5}\\ =12.8\end{array}$

Therefore, the area under the curve is 12.8.