AP Calculus AB / Unit 6: Integration and Accumulation of Change
6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
Notes
The definite integral is the limit of Riemann sums:
$$\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{i=1}^n f(x_i)\,\Delta x,\quad \Delta x=\frac{b-a}{n}$$
Practice
$\displaystyle \lim_{n\to\infty}\sum_{i=1}^n\left(\frac{2i}{n}\right)^3\frac{2}{n}$ equals:
$\displaystyle \Delta x=\frac{2}{n}$ and $\displaystyle x_i=\frac{2i}{n}$ run from 0 to 2.
In $\displaystyle \int_a^b f(x)\,dx$, the $dx$ comes from:
It's the limit of the widths.
$\displaystyle \sum_{i=1}^4 i^2=$
$1+4+9+16$.
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