j-james/math

Logo

All my math notes, now in Markdown.

View the Project on GitHub j-james/math

5-2 Riemann Sums and Definite Integrals

Learning Targets

You will be able to

Concepts / Definitions

Riemann Sums

Definition of a Riemann Sum

Example of a Riemann Sum

Note: The area will be net area! Example:

Net Area under the curve

Definition of the Definite Integral

If $f$ is defined on the closed interval $[a, b]$ and the limit of a Riemann sum of $f$ exists, then we say $f$ is integrable on $[a, b]$ and we denote the limit by

\[\lim_{\Delta x\to 0} \sum_{i=1}^{n}f(c_i)\Delta x_i = \int_a^b f(x)dx\]

The limit is called the definite integral of $f$ from $a$ to $b$. The number $a$ is the lower limit of integration, and the number $b$ is the upper limit of integration.

\[\lim_{n\to\infty} \sum_{i=1}^n f(x_i)\Delta x\]

Parts of the Definite Integral

Note: The answer must be a number, e.g. your answer cannot be $\int_a^b f(x)dx = \infty$. Additionally, this means that the variable of integration does not matter (and not $a$ or $b$.)

$\int_a^b f(x)dx = \int_a^b f(t)dt = \int_a^b f(u)du$

Definitions

Displacement

\[Displacement\ = \int_a^b (velocity)\ dt\]

Distance

\[Distance\ = \int_a^b \lvert\ velocity\ \rvert\ dt\]