SUNY Geneseo Department of Mathematics

Definite Integral Discussion

Math 221 02
Fall 2020
Prof. Doug Baldwin

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

Definite integrals are an application of Riemann sums, described at length in section 5.2 of our textbook. This discussion starts to explore, and gives you a chance to practice using, the consequences of definite integrals being sums.

The colloquial definition of the definite integral is “the definite integral of f(x) is the area under the graph of f(x).” In what ways in this definition wrong or incomplete?

Here are the values of a few definite integrals:

\[\int_0^1 x^2\,dx = \frac{1}{3}\]
\[\int_0^2 x^2\,dx = \frac{8}{3}\]
\[\int_0^2 x\,dx = 2\]

The integral from 0 to 2 of x² is done as a Riemann sum in our textbook. Can you confirm either of the others for yourself using Riemann sums?

Based on these values, what are...

\[\int_1^2 x^2\,dx\] \[\int_0^2 3x^2\,dx\] \[\int_0^2 x^2 + x\,dx\]

Without using Riemann sums, calculators, integration rules you might already know, etc, what do you expect

\[\int_0^{2\pi} \sin x\,dx\]

is? Hint: look at the graph of sin x over this interval.