Riemann Sum

• The area under a curve
• we split it up into $n$ rectangles
  • Find $\mathrm{\Delta }x=\frac{b-a}{n}$
  • Discretize: $x_i=\mathrm{\Delta }xi+a$, and $x_0=a,x_n=b$
  • It is easier to take the height of the rectangle to be the right end point
  • $A_i=f(x_i)\mathrm{\Delta }x$
  • If we take $\underset{n\to \mathrm{\infty }}{lim}$, then we get the exact area
  • $A=\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^n{A}_i=\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^{n}f(x_i)\mathrm{\Delta }x$
• Remember rules of summation, such as $\sum _{i=1}^{n}1=n$ and $\sum _{i=1}^{n}i=\frac{n(n+1)}{2}$
  • math117: you will be given these