Q. 4 - Calculus

Question

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible.

• the geometric interpretations of the n-rectangle trapezoid sum for a function f on an interval $[a, b]$
  

Step-by-Step Solution

Answer

Ans: The trapezoid sum is given by ${\int }_a^{b}f\left(x\right)dx=\frac{\Delta x}{2}\left[f\left(x_0\right)+2f\left(x_1\right)+\dots \dots .+2f\left(x_{n-1}\right)+f\left(x_n\right)\right]$

Step 1. Given information:

The geometrical interpretation of n rectangle trapezoid sum of the function f over an interval [a,b].

Step 2. Finding the trapezoid sum:

Little trapezoids.

Let f is continuous and defined on an interval

$[a, b]$, then interval $[a, b]$is divided into n number of subintervals of equal length $\Delta x$ $a=x_0<x_1<\dots ..<x_n=b$

The trapezoid sum is given by

${\int }_a^{b}f\left(x\right)dx=\frac{\Delta x}{2}\left[f\left(x_0\right)+2f\left(x_1\right)+\dots \dots .+2f\left(x_{n-1}\right)+f\left(x_n\right)\right]$