Q. 9 - Calculus

Question

Notation: Describe the meanings of each of the following mathematical expressions or how they are commonly used in this chapter:      

${\int }_a^{b}f\left(x\right)dx$

Step-by-Step Solution

Answer

Let f be a function defined on an interval [a, b]. The definite integral of f from $x=a$ to $x=b$ is defined to be the number,

${\int }_a^{b}f\left(x\right)dx=\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left(x_k^{*}\right)\Delta x$

if this limit exists.

Step 1. Given information.

We are given an expression, 

${\int }_a^{b}f\left(x\right)dx$

Step 2. Defining the expression

Let f be a function defined on an interval [a, b]. The definite integral of f from $x=a$ to $x=b$ is defined to be the number,

${\int }_a^{b}f\left(x\right)dx=\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left(x_k^{*}\right)\Delta x$

if this limit exists, where $\Delta x=\frac{b-a}{n},x_k=a+k\Delta x$, and $x_k^{*}$ is any choice of point in $\left[x_{k-1},x_k\right]$.