5.4 - Indefinite Integrals and the Net Change Theorem

We got our review quiz back, and I had a 77 on it. Looking back over it, it was mainly careless errors and forgetting my trig identities, so it's annoying but not devastating. Anyway, on to 5.4

5.3 took a lot of mental effort to process, but, from what I've read so far, 5.4 is just some applications of it, so it shouldn't be quite as taxing. That's a good thing, because I have a quiz to do after the homework for this section. Fun times.

To make it easier to write down, we're using ⌠f(x)dx as the notation for an antiderivative of f(x). Since the integral on a segment is the definite antiderivative, using the integral without a given segment is an indefinite integral, and will be a general antiderivative with a +C. If you have a given segment, the definite integral gives a number - an absolute area. If you use the indefinite integral, you get a function.
We then have a table of indefinite integral formulas, which are pretty much the opposite of the derivative formulas + C, so I'm not going to bother summarizing them.

Also, the indefinite integral is only valid on an interval. It compares it to the general antiderivative, but I don't fully understand the difference it's showing. Something to ask about later.


Now, since ⌠f(x)dx is the antiderivative of the function, then ⌠F'(x) is the derivative of the antiderivative of function, so it's just another way of saying that function. But, looking at it as the derivative of the antiderivative, it becomes the rate that the antiderivative changes as x changes.
So stating our our ⌠f(x)dx = F(b)-F(a) as ⌠F'(x)dx = F(b)-F(a) makes its use in science applications more apparent, since we know that the rate the integral changes is the same as the net change between the interval.

Example: if C(x) is the cost of producing x units of something, then we know the marginal cost (C'(x)) is the rate of change in the cost. So ⌠C'(x)dx = C(x2)-C(x1) would give us the increase in cost from one amount to the other.

There are a few more examples, but that's pretty much all their is to this chapter. A littler easier to wrap my head around it.