Chapter 11.1, Problem 1E

To determine

RUN the program to approximate the area of the shaded region.

Answer to Problem 1E

0.33835

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).

The program for calculating the area of shaded region using 10 rectangles :

  $\begin{array}{l}10\text{ LET }X=0.1\\ 20\text{ FOR }N=1\text{ TO 10}\\ \text{30 LET }Y=X\uparrow 2\\ 40\text{ LET }A=A+Y*0.1\\ 50\text{ LET }X=X+0.1\\ 60\text{ NEXT }N\\ 70\text{ PRINT "AREA IS APPROXIMATELY"; }A\\ 80\text{ END}\end{array}$

Calculation:

We can approximate the area of the shaded region by drawing 100 rectangles having base vertices at x = 0 , 0.01 , 0.02, 0.03,…,1.00 , and computing the sum of the areas of the 100 rectangles.The base of each rectangle is 0.01 , and the height of each rectangle is given by $y=x^2$ .

The following comuter program will comute and add the areas of the 100 rectangles .In line 30, Y is the height of each rectangle.In line 40, A gives the current total of all the areas.

  $\begin{array}{l}10\text{ LET }X=0.01\\ 20\text{ FOR }N=1\text{ TO 100}\\ \text{30 LET }Y=X\uparrow 2\\ 40\text{ LET }A=A+Y*0.01\\ 50\text{ LET }X=X+0.01\\ 60\text{ NEXT }N\\ 70\text{ PRINT "AREA IS APPROXIMATELY"; }A\\ 80\text{ END}\end{array}$

If the program is run , the computer will print :

AREA IS APPROXIMATELY 0.33835