Chapter 11.1, Problem 3E

To determine

Find whether the actual area of the shaded region is more of less than the value given by computer program.

Answer to Problem 3E

Less

Explanation of Solution

Given:

The shaded region is bounded by the graph of $y=x^2$ , by the x -axis , and by the vertical line through the points (1,0) and (1,1).

Calculation:

Computer Program:

We can approximate the area of the shaded region by drawing 1000 rectangles having base vertices at x = 0 , 0.001 , 0.002, 0.003,…,1.000 , and computing the sum of the areas of the 1000 rectangles.The base of each rectangle is 0.001 , and the height of each rectangle is given by $y=x^2$ .

The following comuter program will comute and add the areas of the 1000 rectangles .In line 30, Y is the height of each rectangle.In line 40, A gives the current total of all the areas.

  $\begin{array}{l}10\text{ LET }X=0.001\\ 20\text{ FOR }N=1\text{ TO 1000}\\ \text{30 LET }Y=X\uparrow 2\\ 40\text{ LET }A=A+Y*0.001\\ 50\text{ LET }X=X+0.001\\ 60\text{ NEXT }N\\ 70\text{ PRINT "AREA IS APPROXIMATELY"; }A\\ 80\text{ END}\end{array}$

If the program is run , the computer will print :

AREA IS APPROXIMATELY 0.333833

Actual Calculation:

We find the area of the shaded region by integration:

  $\begin{array}{l}A={\displaystyle \underset{0}{\overset{1}{\int }}{x}^{2}dx}\\ \Rightarrow A={{\displaystyle \left[\frac{x^3}{3}\right]}}_0^1\\ \Rightarrow A=\frac{1^3}{3}-\frac{0^3}{3}\\ \Rightarrow A=\frac{1}{3}=0.\overline{3}<0.333833\end{array}$

So, the actual area of the shaded region is less than the value computed by the computer program.