Q2. - Geometry

Question

Modify the given program so that it will use 1000 smaller rectangles with base vertices at$0,0.001,0.002,0.003,\mathrm{........},1.000$ to approximate the area of the shaded region. RUN the program.

Step-by-Step Solution

Answer

Area is approximately$0.333833$ .

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  rectangles having base vertices at$x=0,0.001,0.002,0.003,\dots ,1.000,$and computing the sum of the areas of the1000  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 line30Y is the height of each rectangle.In line 40A, A gives the current total of all the areas.

Step 3. Do the calculation.

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

Therefore, If the program is run, the computer will print AREA IS APPROXIAMTELY $0.333833$