- This section formalizes our intuition from the preceding section:
- First of all, we can't expect our function to be positive all the time, so
we define "signed area":
and then we give a name to this area:
- The area can be calculated by chopping the interval [a,b]
up into smaller and smaller chunks, and then using rectangular approximations
to the small areas.
- In our work so far, the intervals into which the interval
is chopped have been equal-sized: no reason to stick with that.
where P is a partition of the interval [a,b]:
- If the size of the intervals goes to zero, then the
approximations get better and better, until they're perfect!
where is the width of the largest subinterval, for the partition.
- In our work so far, the intervals into which the interval
is chopped have been equal-sized: no reason to stick with that.
- Which functions are "integrable"?
- Here's how we'd rewrite our generalization that
Area under the graph of f from a to b = : This is a preview of section 6.2, p. 315 -- the Mean Value Theorem for Integrals. A couple of theorems from this section fall right out based on this idea:
From which we can derive the following bounds on an integral: if , then
- A couple of others don't, but are relatively intuitive even so:
- First of all, we can't expect our function to be positive all the time, so
we define "signed area":
- Examples:
- #12, p. 270
- #13, p. 270
- #25, p. 271
- #60, p. 272
- #64, p. 272
- Well folks, here it is:
It says that to integrate, all you have to do is think of an antiderivative F of the integrand f.
- Let's derive/motivate the FTC using the linearization (again! wow,
it's useful...)
- Examples:
- #12, p. 276
- #40, p. 277
- #49, p. 277