Q. 1 - Calculus

Question

State the definition of the definite integral of an integrable function f on [a, b].

Step-by-Step Solution

Answer

If f is a function defined on an interval $[a,b]$ then the definite integral of f from $x=a$ to $x=b$ is defined by a number that follows:

$$\begin{gathered} {\int }_a^{b}f\left(x\right) dx=\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left({\stackrel{*}{x}}_k\right) ∆x \\ \mathrm{Where} ∆\mathrm{x}=\frac{\mathrm{b}-\mathrm{a}}{\mathrm{n}} \\ {\mathrm{x}}_{\mathrm{k}}=\mathrm{a}+\mathrm{k}∆\mathrm{x} \end{gathered}$$

Step 1. Given Information.

The function $f$ on $[a,b]$

Step 2. Definition of definite integrals.

If f is a function defined on an interval $[a,b]$ then the definite integral of f from $x=a$ to $x=b$ is defined by a number that follows:

$$\begin{gathered} {\int }_a^{b}f\left(x\right) dx=\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left({\stackrel{*}{x}}_k\right) ∆x \\ \mathrm{Where} ∆\mathrm{x}=\frac{\mathrm{b}-\mathrm{a}}{\mathrm{n}} \\ {\mathrm{x}}_{\mathrm{k}}=\mathrm{a}+\mathrm{k}∆\mathrm{x} \end{gathered}$$