- Next exam: Tuesday, 4/20
- Midterm grades: see me if you wish to know it (and my apologies that they didn't make it on-line -- I thought that they'd been registered, but evidently not).
- Sorry: I haven't graded your 4.6 homework yet.
-
Thu 4/8 Section 5.2 Read 5.3 for Monday; Homework for 5.2: pp. 270-272, #7, 10, 11, 14, 15, 16, 26, 63, 69, 74, 75, 77 (to hand in -- due 4/13).
- Dirt!
- Suppose that you clock your speed at each hour along a trip, in
mph:
Estimate the distance travelled:hour 0 1 2 3 4 5 6 7 8 9 10 speed (mph) 0 30 45 30 70 65 70 70 45 30 35
- Assumes continuity of underlying speed function
- We might fit straight line segments between data points: this gives rise to the so-called "Trapezoidal rule".
- Suppose that you clock your speed at each hour along a trip, in
mph:
- Other useful rules are rectangular: Left/Right/Midpoint rules:
- relax continuity! Use rectangles...
- also called rectangle rules: lrr, rrr
- Estimates:
- RRR: d = 45.5 mph * 10 hours = 455 miles
- LRR: d = 49mph * 10 hours = 490 miles
- Trapezoidal rule = 47.25 mph * 10 hours = 472.5 miles
- Notes:
-
- Whenever you have two estimates, you have a third:
- the trapezoidal rule is the average of left and right rectangle rules.
- Each of these estimates corresponds to computing a rectangle height (an
average speed), and then multiplying by the total time (10
hours). Hence we're using the "dirt" formula, and computing the
average (the "area under the curve", even though there's no curve!) in
several different ways.
- Basically, every function can be used to create a table like this, and
then we can use these strategies.
- Often (as in the table above) we have data located at equal intervals (
). When you have a curve (or a formula) for f, you can take the time intervals shorter and shorter, until you get as close to the area as you need:
(notice that
that is, that
)
Then, in fact,
- Examples:
- #5, p. 258 (also trapezoidal) -- illustrates the problem of using a midpoint method with a table.
- #8, p. 258 (also M3)
- #9, p. 258 -- just create a table, and away you go....
- Generalizations?
- The area can be calculated by finding an average value on the
interval, times the length of the interval:
Area under the graph of f from a to b = where bywe mean an average value of f.
- 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.
- If the size of the intervals goes to zero, then the approximations get better and better, until they're perfect!
- The area can be calculated by finding an average value on the
interval, times the length of the interval:
- 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.
- 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]:
such that
- If the size of the intervals goes to zero, then the
approximations get better and better, until they're perfect!
where
- In our work so far, the intervals into which the interval
is chopped have been equal-sized: no reason to stick with that.
- Properties of the Definite Integral:
- 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.
- 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