- Once again we'll be in here on Friday, hopefully working on 4.3.
- Your 3.7 homework is not yet returned: graded 13, 14, 23.
- Your 3.9 homework is due.
- Let's take a look at this Handout, and Stewart's animations.
- #6
- #15
- This section generalizes our intuition and examples 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.
If the size of the intervals goes to zero, then the approximations get better and better, until they're perfect!
Assuming that we use equal-sized subintervals, the definite integral of $f$ from $a$ to $b$ is defined by \[ \int_{a}^{b}f(x)dx = \lim_{n \to \infty}\sum_{i=1}^{n}f(x_i^*)\Delta x \] where the $x_i^*$ are sample points, one from each of the $n$ rectangles of width $\Delta x = \frac{b-a}{n}$.
Theorem 3, p. 297, tells us that this integral will be defined ($f$ is integrable) whenever $f$ is continuous (or has at most a finite set of discontinuities).
Theorem 4, p. 298, tells us that we can take RRR as our method and be assured of getting the right answer. But we could just as well use the LRR, or midpoint, or trapezoidal, etc.
- Properties of the Definite Integral:
- Here's how we'd rewrite our generalization that
Area under the graph of f from a to b =
width of the interval times the average value =
$(b-a)\bar{f}$
where by
$\bar{f}\mbox{}$
we mean an average value of f. So
$\int_{a}^{b}f(x)dx=\bar{f}(b-a)$
- Examples:
- #7, p. 306
- #10
- #21 (see p. 298 for the equations you might need, and example 2)
- #33
- First of all, we can't expect our function to be positive all the time, so
we define "signed area":