Q1. - Geometry

Question

A better approximation can be found by using 100  smaller rectangles with base vertices at $0,0.01,0.02,0.03,\mathrm{........},1.00$ . Change lines$10,20,40$  and 50 as follows:$\begin{array}{l}{10}^{}LE{T}^{}X=0.01\\ {20}^{}FO{R}^{}N=1^{}T{O}^{}100\\ {40}^{}LE{T}^{}A=A+Y*0.01\\ {50}^{}LE{T}^{}X=X+0.01\end{array}$

                                       

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

Step-by-Step Solution

Answer

Area is approximately $0.33835$ .

Step 1. Write the given information.

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

 

Step 2. Concept Used.

We can approximate the area of the shaded region by drawing 100  rectangles having base vertices at $x=0,0.01,0.02,0.03,\dots ,1.00,$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.

Step 3. Do the calculation.

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

Therefore, If the program is run, the computer will print AREA IS APPROXIAMTELY  $0.33835$