Question

In Archimedes's method for finding the area of a curve(function), we divide the area into intervals of width ‘1’ and construct rectangles of width ‘1’ and height as the value of the function on the rightmost point of the interval.

• A. True
• B. False

Answer 1

Answer: (B)

Let’s say we want to find the area of a function from x = 0 to x=b between the curve and x-axis.

In Archimedes’s method for finding the area of a curve, we will divide the given curve into n intervals of width $\frac{b-0}{n}.$ In each In each interval we will construct a rectangle with width as the length of interval and height as the value of the function on the rightmost point of the interval(when we approach from top). We will now calculate the area of the rectangle formed. For example, if we consider the interval $[\frac{b}{n},\frac{2b}{n}],$ corresponding to the second rectangle, the height of the rectangle will be $f\left(\frac{2b}{n}\right).$ Similarly the height of $k^{th}$ rectangle will be $f\left(\frac{kb}{n}\right)$.

But in the question it is given that we divide the area into rectangles of unit width. We will be initially dividing the area into n intervals and later we apply the limit n tends to infinity to find the exact area. So the width of the intervals will tend to zero as n tends to infinity. So the given statement is false.