- Assignment 8.2 due
- Exam is not yet graded - how did it go?
- What kinds of problems are we looking at in this section? Arc
length and surface area calculations for areas of revolution:
- Let's start with arc length:
The usual trick is to discretize something continuous, and then pass to a limit:
Now hopefully we can figure out how the formula will look:
So what's the general formula?
- For surface area (for areas of revolution)
We discretize bits of the surface, and then make a series of little bands; take the limit, and get the general formula:
- Examples:
- #5, p. 475
- #12, p. 475
- #39, p. 476
- #51, p. 477
- You've already solved a differential equation:
The solution is
-- so there's an unknown constant k (in this case, it corresponds to the "initial value"). If we tell what y equals at t=0, then we'll get the particular solution, (and that's often very important, that we get the exact solution and unique solution given the data).
We know that exponential functions have the property that the derivative is proportional to its function value (and
is the special function for which the constant of proportionality is exactly 1).
- As indicated on page 505, differential equations can be a little different:
or
The first doesn't involve y, and the second involves the second derivative of y.
- The general method for solving differential equation is to stare at them until
the solution comes to you!;): what are solutions to the two
differential equations above?
- Today we'll do a lab in which we consider "Newton's Law of Cooling". Before we
start, however, let's make up a few differential equations of our own: consider
the following functions, and figure out differential equations that they solve:
-
- Here's one place: Newton's Law of Cooling (Lab)
- and