Friday April 4 and Monday April 7

4/4.
5.2. When a function has a different formula on different intervals, you're better off integrating it separately on the different intervals and adding them together.

5.3,4. The Fundamental Theorem of Calculus, in two versions:
v2. if you integrate something continuous then differentiate, you get back the function you started with.
v1. if you differentiate something differentiable then integrate, you almost get back the function you started with; you get it up to an additive constant.

An example of an integral to compute total food eaten by a pet hamster and lion.

An example with the Baby Name Wizard of computing the number of Allens between the ages of 30 and 40, as an integral. I drew the graphs of Beulah, Brianna, Adolf, and Monica.

4/7.
Definite integrals vs. indefinite. Definite integrals are numbers; indefinite integrals are lazily written (by everyone) as functions, when really they're "functions up to an additive constant". So make sure you include that "+ C" when indefinitely integrating something.

5.6. Recall that the chain rule of differentiation (f(g(x)))' produces a product f'(g(x)) g'(x). So when faced with an integral of a product, see if you can recognize the product as being the derivative of a composite function.

Why is the dx so important to remember in integrals? Because you need to change it, too, when you do change of variable. Otherwise you get the wrong answer, as I demonstrated.