Q3. - Geometry

Question

Is the actual area of the shaded region more or less than the value  given by computer program? Explain

Step-by-Step Solution

Answer

The actual area of the shaded region is less than the value computed  by the computer  program

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. Calculate the area using computer program.

We can approximate the area of the shaded region by drawing 1000 rectangles having base vertices a$x=0,0.001,0.002,0.003,\dots ,1.000,$tand 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 line40 , 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 APPROXIAMTELY  $0.333833$.

Step 3. Calculate the Actual area.

We find the area of the shaded region by integration:

$\begin{array}{l}A=\underset{0}{\overset{1}{\int }}{x}^{2}dx\\ \Rightarrow A={\left[\frac{x^3}{3}\right]}_0^1\\ \Rightarrow A=\frac{1^3}{3}-\frac{0^3}{3}\\ \Rightarrow A=\frac{1}{3}=0.\overline{3}<0.333833\end{array}$

So, the actual area of the shaded region is less than the value computed by the computer program.