Overview
Even though we now know many different techniques for integration, there are some functions whose antiderivative simply cannot be written as an elementary function, for example \(\int e^{x^2}\; dx\). Nevertheless, we often need to find definite integrals for such functions, and we therefore need to discuss what to do in such cases.
There are two main approaches:

We can use “Numerical Integration” techniques to make numerical estimates that are quite close to the real value of the wanted integral: We already know Left, Right, and Midpoint Riemann Sum Rules, and we will briefly discuss two more techniques that are somewhat more accurate.

We can replace the troublesome integrand by an appropriate Taylor or Maclaurin series representation, and then integrate this series term by term.
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

(Review, Section 4.2) State the definition of a Riemann Sum and draw graphs with boxes that represent the Left, Right, and Midpoint sum of a given function.

Given a definite integral, draw a picture that represents the estimate for the integral made by the Trapezoid Rule.

Given a definite integral, estimate its value by using the Trapezoid Rule.

State the types of functions for which the Midpoint and Trapezoid Rule are an over or underestimate.
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

State the graphical meaning of Simpson’s Rule.

Given a definite integral, draw a picture representing the estimate made by Simpson’s Rule and calculate this estimate’s value using Simpson’s rule.
To prepare for class

Read all of Section 5.6 in Active Calculus.

Watch the following videos (the first one is a review, the other two are new material):

Read subsection 8.6.2: Manipulating Power Series, with special focus on the Power Series Differentiation and Integration Theorem.

Watch the following video (watch the whole video as a review, but take special note of the last part, starting at 5:33):

Do the Preview Activity for this section (on WeBWorK if required by your teacher).