Day 15, Mat227

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Announcements:
- Your exams are returned. It was kind of feast or famine:
  - The exam
  - A key
  - As a curve, you can add 3 points to your score (out of 85).
- For those of you who bombed, the final will be in two parts -- new stuff and old stuff. If your "old stuff" score is higher than your lowest midterm exam, I'll replace the midterm score with that score. So recovery is possible. That being said, you can't fail again. So you have to ask yourself if you're going to be able to make it through.
- You have a homework due (section 4.5)
- Some of you aren't too happy with Riemann sums. The difference is between producers and consumers:
  - Producers design/create integrals that are used to solve problems (which computers can't do, or do with great difficulty);
  - Consumers merely learn to do what computers can do better.
  Which path offers more job security?
  I want to create producers, not consumers. Producers will understand and be able to use Riemann sums.
Today I'd like to get to 5.2, since we lost a week.
Before we go on, however, we should probably do a few more examples from 5.1:
- #2, p. 349
- #3
- #6
- #17
- #31
- #41
- #45
- #46
Section 5.2: the volume
Let's see how to compute volumes of some simple objects using integrals.
Before you may have memorized some of these formulas. Now you can derive them!

This is the power of calculus.

In every case for what follows, we begin with what I consider the most important formula for integral calculus:

${V=\int{dV}}$
To calculate any quantity ${V}$ , we simply divide it up into infinitesimal quantities ${dV}$ , and add an infinity of them up.
Of course the letter ${V}$ is not crucial -- but in this chapter we consider volumes, and so we use ${V}$ . In the early going, we were computing areas, so one might write the most important formula for integral calculus as

${A=\int{dA}}$
That could be used to compute Andrews as well -- an Andrew is a sum of an infinite number of infinitesimal chunks of Andrew.
If we were computing masses, we'd write

${M=\int{dM}}$
a mass is made up of an infinite numbers of infinitesimal masses.

Now for volumes we usually know the extent of the object, so we'll march along the x-axis from a to b, stop at x, and add in the little ${dV({x})}$ that is found there. So in each case what we need to do is find ${dV({x})}$

${V=\int_{a}^{b}dV(x)}$
- The cube of side length ${s}$ has a differential of volume equal to the differential width ${dx}$ times the area of the face, ${A({x})}$ :
  
  ${dV(x)=A(x)dx=s^2dx}$
  
  ${V=\int_{0}^{s}dV(x)}$
- The right circular cylinder of radius ${r}$ and height ${h}$ has differential volume
  
  ${dV(x)=A(x)dx=\pi{r}^2dx}$
  
  ${V=\int_{0}^{h}dV(x)}$
- The cone of base radius ${r}$ and height ${h}$ , laid along the x-axis as its central axis (and its tip at ${x=h}$ ) -- draw a picture! -- has differential volume
  
  ${dV(x)=A(x)dx=\pi\left(r(1-\frac{x}{h})\right)^2dx=\pi{r^2}(1-\frac{2x}{h}{+}\frac{x^2}{h^2})dx}$
  and then, as usual,
  
  ${V=\int_{0}^{h}dV(x)}$
  (do you see a pattern here? The only thing that changes is the differential -- and the name on the variable at the endpoint. It's all about the differential! And there are those who would drop them -- shame!)
- The pyramid of height ${h}$ , and square base ${s}$ .
  What would its differential look like? How should we go about computing this volume?
- The sphere of radius ${r}$
  
  ${dV(x)=A(x)dx=\pi\left(\sqrt{r^2-x^2}\right)^2dx=\pi(r^2-x^2)dx}$
  
  ${V{=}2\int_{0}^{r}dV(x)}$
  (the "2" is out in front because we're using symmetry: we calculate the volume of half the sphere in running from 0 to r, then double it).
- Examples:
  - #48, p. 361
  - #52
  - #4 and 5, p. 360

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