- Your homework is not returned yet -- sorry. Graded 1, 8, 12.
- #15:
- sharp
- ebert
- My attempt -- I think that I had the indices wrong for $x_{ij}$ and $y_{ij}$....
- You have a homework due (section 15.2).
The author makes a great deal of "Type I" (p. 1013) and "Type II" (p. 1014) regions. This is a mere convenience: each has a pair of "rectangular" edges, in either $x$ or $y$ directions. Turn your book sideways.
It's no big deal to turn a general region into such regions -- the question is whether you need to, and how, if so.
The essential point is this: we're really trying to do is parameterize the boundaries in a sensible way, and, if we can, then we can go back to Fubini's theorem and do twice as much work as we did when we did univariate integrals.
Take a look at Figure 18, p. 1018. We can break it down into a pair of these types of regions, and move ahead....
There are a few properties of integrals worth paying attention to in this section, such as the inequality at the bottom of 1018. These should make intuitive sense to you -- hopefully so!
- Let's draw the regions being integrated over for problems #1-6
- #6
- #15
- #17
- Let's start by taking a look at Example 2, p. 1024, to identify
the problem(s):
- Changing to polar coordinates isn't too bad: we just switch from
describing our region using Cartesian coordinates $x$ and $y$, and move
to polar coordinates $r$ and $\theta$, via the transformation equations
\[
r^2=x^2+y^2 \hspace{1in}
x=r\cos{\theta} \hspace{1in}
y=r\sin{\theta}
\]
For "Polar rectangle" $R=\left\{ (r,\theta)\
|\ a\le{r}\le{b},\ \alpha\le\theta\le\beta\right\}$, substitution gives
\[
\int\!\!\!\int_{R}f(x,y)dA =
\int_{\alpha}^{\beta}\int_{a}^{b}f(r\cos(\theta),r\sin(\theta))r dr d\theta
\]
Notice that the differential of area, $dA$, has been replaced by $r dr d\theta$: take a look at Figures 3, 4, and 5, p. 1022/23:
- Examples:
- Problems 1-4, p. 1026
- #5 (why does the integral represent the area?)
- #10
- #14 (a little more complicated, and we'll just set it up so that you can practice your univariate integrals!:)
- #36 (and the important question: how should the nozzles be designed so that you deliver a constant amount of water per hour at every point in the field? Wouldn't that make sense?)