Chapter 11.3, Problem 2E

To determine

Modify the given computer program so that it will use 1000 trapezoids with base vertices at 1.000, 1.001,1.002,…,2.000 to approximate the area of the shaded region.RUN the program and find the answer.

Answer to Problem 2E

0.3333334

Explanation of Solution

Given:

The shaded region is bounded by the graph of $y=x^2$ , by the x -axis , and by the vertical lines x = 1 and x = 2.

The program for computing and adding the areas of 5 trapezoids :

  $\begin{array}{l}10\text{ LET }X=1\\ 20\text{ FOR }N=1\text{ TO 5}\\ \text{30 LET B1}=X\uparrow 2\\ \text{40 LET B2}=\left(X+0.2\right)\uparrow 2\\ 50\text{ LET }A=A+0.5*0.2*\left(B1+B2\right)\\ 60\text{ LET }X=X+0.2\\ 70\text{ NEXT }N\\ 80\text{ PRINT "AREA IS APPROXIMATELY"; }A\\ 90\text{ END}\end{array}$

Calculation:

A better approximation can be found by using 100 smaller trapezoids with base vertices at 1.000,1.001,1.002,…1.999,2.000 , and computing the sum of the areas of the 1000 trapezoids.The parallel bases are vertical segments from the x -axis to the curve $y=x^2$ .The altitude is a horizontal segment with length 0.001.

The area of the shaded region is approximated by the sum of the areas of the 1000 trapezoids.

The following comuter program will comute and add the areas of the 1000 trapezoids .In line 30 and 40, B1 and B2 are the parallel bases of the trapezoid.In line 50, A gives the current total of all the areas.

  $\begin{array}{l}10\text{ LET }X=1\\ 20\text{ FOR }N=1\text{ TO 1000}\\ \text{30 LET B1}=X\uparrow 2\\ \text{40 LET B2}=\left(X+0.001\right)\uparrow 2\\ 50\text{ LET }A=A+0.5*0.001*\left(B1+B2\right)\\ 60\text{ LET }X=X+0.001\\ 70\text{ NEXT }N\\ 80\text{ PRINT "AREA IS APPROXIMATELY"; }A\\ 90\text{ END}\end{array}$

If the program is run , the computer will print :

AREA IS APPROXIMATELY 2.3333334