Q. 51 - Calculus

Question

Use Definition 4.6 and the definition of increasing to prove that if a function f is positive and increasing on $[a, b]$, then the left sum with n rectangles is less than or equal to the right sum with n rectangles.

Step-by-Step Solution

Answer

It is proved that if a function f is positive and increasing on $[a,b]$, then the left sum with n rectangles is less than or equal to the right sum with n rectangles.

Step 1. Given Information

We are given that a function f is positive and increasing.

Step 2. Proving the statement

The left-sum defined for n rectangles on $[a, b]$ is $\sum _{k=1}^{n}f\left(x_{k-1}\right)\Delta x$.

Where, $\Delta x=\frac{b-a}{n},x_k=a+k\Delta x$.

The right sum defined for n rectangles on $[a, b]$ is $\sum _{k=1}^{n}f\left(x_k\right)\Delta x$.

Since f is increasing,

$x_{k-1}\le x_k$

Therefore, $\sum _{k=1}^{n}f\left(x_{k-1}\right)\Delta x\le \sum _{k=1}^{n}f\left(x_k\right)\Delta x$.