Chapter 11.3, Problem 3E

To determine

Modify the computer program so that it will use ten trapezoids to approximate the area of the shaded region.RUN the program.

Compare your answer with that obtained when ten rectangles are used to compute the area of the shaded region.

Answer to Problem 3E

Answer by 10 trapezoid program is 0.334 , which is closer to actual area of shaded region $0.\overline{3}$

Answer by 10 rectangle program is 0.385

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 = 0 and x = 1.

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

  $\begin{array}{l}10\text{ LET }X=0\\ 20\text{ FOR }N=1\text{ TO 5}\\ \text{30 LET B1}=X\uparrow 2\\ \text{40 LET B2}=\left(X+0.2\right)\uparrow 2\\ 50\text{ LET }A=A+0.5*0.2*\left(B1+B2\right)\\ 60\text{ LET }X=X+0.2\\ 70\text{ NEXT }N\\ 80\text{ PRINT "AREA IS APPROXIMATELY"; }A\\ 90\text{ END}\end{array}$

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:

A better approximation can be found by using 10 smaller trapezoids with base vertices at 0.0,0.1,0.2,…0.9,1.0 , and computing the sum of the areas of the 10 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 0.1.

The area of the shaded region is approximated by the sum of the areas of the 10 trapezoids.

The following comuter program will comute and add the areas of the 10 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.

  $\begin{array}{l}10\text{ LET }X=0\\ 20\text{ FOR }N=1\text{ TO 10}\\ \text{30 LET B1}=X\uparrow 2\\ \text{40 LET B2}=\left(X+0.1\right)\uparrow 2\\ 50\text{ LET }A=A+0.5*0.1*\left(B1+B2\right)\\ 60\text{ LET }X=X+0.1\\ 70\text{ NEXT }N\\ 80\text{ PRINT "AREA IS APPROXIMATELY"; }A\\ 90\text{ END}\end{array}$

If the program is run , the computer will print :

AREA IS APPROXIMATELY 0.334

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

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

If the program is run , the computer will print:

AREA IS APPROXIMATELY 0.385

The actual area of the shaded region is $0.\overline{3}=\mathrm{0.3333333333...}$

So, clearly , the computer program that calculates the area of the shaded region via 10 trapezoids is approximately closer to the actual answer.