Area

Summation Formulas

$$\sum_{i=1}^n ka_i=k\sum_{i=1}^n a_i \sum_{i=1}^n (a_i\pm b_i)=\sum_{i=1}^n a_i \pm \sum_{i=1}^n b_i$$

$$\sum_{i=1}^n c=cn \sum_{i=1}^n i=\frac{n(n+1)}{2}$$

$$\sum_{i=1}^n i^2=\frac{n(n+1)(2n+1)}{6} \sum_{i=1}^n i^3=\frac{n^2(n+1)^2}{4}$$

Area of a Plane Region

If $f$ is continuous and nonnegative on the interval $[a,b],$ the area of the region bounded by the graph of $f,$ the $x$ -axis, and the vertical lines $x=a$ and $x=b$ is

   Area$$\equiv \lim_{n\to\infty}\sum_{i=1}^n f(c_i) \Delta x,\qquad x_{i-1}\le c_i\le x_i$$

where $\Delta x=(b-a)/n.$