Lesson 22: Approximating Areas

• Sigma notation of the form $\displaystyle\sum_{i=1}^{n}a_i$ is useful to express long sums of values in compact form. 
• Riemann Sums are expressions of the form $\displaystyle\sum_{i=1}^{n}f(x^*_i)\Delta x$, and can be used to estimate the area under the curve $y=f(x)$.
• Approximate the area under the curve $y=f(x)$ on the interval  $[a, b]$ :
  1. Divide the interval  $[a, b]$  into  $n$  subintervals with equal length $\Delta x=\dfrac{b-a}{n}$;
  2. Let $x_i=a+i\cdot\Delta x$;
  3. We can either use Left Endpoint Approximation:  

$A\approx L_n=f(x_0)\Delta x+f(x_1)\Delta x+\cdots+f(x_{n-1})\Delta x=\displaystyle\sum_{i=1}^{n}f(x_{i-1})\Delta x$

or Right Endpoint Approximation: 

$A\approx R_n=f(x_1)\Delta x+f(x_2)\Delta x+\cdots+f(x_{n})\Delta x=\displaystyle\sum_{i=1}^{n}f(x_i)\Delta x$