Chapter 6.1, Problem 9E

To determine

To calculate: Use Rectangular Approximation Method to estimate the area of the region enclosed between the graph of $f$ and the $x$ - axis for $a\le x\le b$.

Answer to Problem 9E

23.875

Explanation of Solution

Given information:

  $f(x)=x^2-x+3,a=0,b=3$

Calculation:

Let’s divide the summation in three equal parts,

  $\therefore \Delta x=\frac{b-a}{n}=\frac{3-0}{3}=1$

Now, our graph is divided into three rectangles, we have to find the areas of these three rectangles for approximate areas.

Now, lets first find out the midpoints of all these rectangles,

  $\therefore x_1=0.5,x_2=1.5,x_3=2.5$ , corresponding values of function are, $f_1=2.75,f_2=3.75,f_3=6.75$

Therefore, Area under the graph is,

  $A=f_1{x}_1+f_2{x}_2+f_3{x}_3=1.375+5.625+16.875=23.875$