Chapter 11.1, Problem 2E

To determine

Modify the program so that it will use 1000 rectangles with base vertices at 0,0.001,0.002,0.003,…,1.000 to approximate the area of the shaded region.RUN the program and find the answer.

Answer to Problem 2E

0.333833

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 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