Chapter 13, Problem 43RE

To determine

To find the area of the shaded region under the graph of the given function by using the indicated

Answer to Problem 43RE

The area of the shaded region $3.825$

Explanation of Solution

Given information:

Consider the given information,

  $f\left(x\right)=\sqrt{x}$

CALCULATION:

Approximate area of the rectangles,

  $\begin{array}{l}\text{Width =}\frac{3-0}{6}=\frac{1}{2}\\ \therefore \text{ Area }=A_1+A_2+A_3+A_4+A_5+A_6\\ =\frac{1}{2}f\left(\frac{1}{2}\right)+\frac{1}{2}f\left(1\right)+\frac{1}{2}f\left(1+\frac{1}{2}\right)+\frac{1}{2}f\left(2\right)+\frac{1}{2}f\left(2+\frac{1}{2}\right)+\frac{1}{2}f\left(3\right)\\ =\frac{1}{2}\left[\sqrt{\frac{1}{2}}+1+\sqrt{\frac{3}{2}}+\sqrt{2}+\sqrt{\frac{5}{2}}+\sqrt{3}\right]\\ =\frac{1}{2}\left[1+\frac{1+\sqrt{3}+2+\sqrt{5}+\sqrt{6}}{\sqrt{2}}\right]\\ =\frac{1}{2}\left[1+\frac{3+\sqrt{3}+\sqrt{5}+\sqrt{6}}{\sqrt{2}}\right]\\ =\frac{1}{2}\left[\frac{\sqrt{2}+3+\sqrt{3}+\sqrt{5}+\sqrt{6}}{\sqrt{2}}\right]\\ =\frac{1}{2}\left[\frac{3+7.83}{\sqrt{2}}\right]\\ =\frac{1}{2}\times 7.83=3.852\end{array}$

Therefore, The area of the shaded region $3.825$