Definite Integral Riemann Sum Definition

Definition (Riemann Sum Definition of Definite Integral)

• Let $a,b\in R$
• Let $a<b$
• Let $[a,b]\subset dom(f)$
• Let $P=\{x_i{\}}_{i=0}^n$ be $a$
• Create a Riemann Partition of $[a,b]$ A function is integrable if: ${\int }_a^{b}f(x)dx={\lim }_{n\to \infty }{\sum }_{i=1}^{n}f(x_i\ast )△x,\forall x_i^{\ast }\in [x_1\dots x_i]$ For Right-Riemann Sum: ${\int }_a^{b}f(x)dx={\lim }_{n\to \infty }{\sum }_{i=1}^{n}f\left(a+(i)\frac{b-a}{n}\right)\left(\frac{b-a}{n}\right),\forall x_i^{\ast }\in [x_1\dots x_i]$ For Left-Riemann Sum: ${\int }_a^{b}f(x)dx={\lim }_{n\to \infty }{\sum }_{i=1}^{n}f\left(a+(i-1)(\frac{b-a}{n})\right)\left(\frac{b-a}{n}\right),\forall x_i^{\ast }\in [x_1\dots x_i]$
• You can create an infinite amount of integrals from a given function, different $a,b,x_i$ values and it will still work

Example to convert Integral form to Riemann sum form

Convert: ${\int }_0^3(x-5)dx$

• $a=0$

• $b=3$

• Note that $[a,b]\subset dom(f)$

• Given $[a,b]=[0,3]$

• $f(x)=x-5$ Then,

• we have $△x\frac{(b-a)}{n}$

• $x_i^{\ast }=a+n△x$