6.3 Arclength. Be able to set up the integral giving the length of the curve traced out by the point with coordinates (x(t), y(t)) between t = a and t = b [Formula 1, Example 1 p.463, Exercises 3, 5, 9]. Special case: be able to set up the integral giving the length of the graph of y = f(x) between x = a and x = b [Formula 2 p. 463, Example 2, Exercises 8,9]. Since these integrals are in general difficult or impossible to calculate analytically (using anti-differentiation) be able to apply approximation methods to estimate them [Exercise 11 - use n=4 if doing by hand].
6.4 Average value. Be able to calculate the average value of the function f(x) on the interval [a,b]: divide the integral by (b-a). [Box p. 468, Example 1, Exercises 5, 7].
6.5 Work. Understand that if the force is a constant F, and displaces
its application point from a to b, the work done is
the product W = F (b-a); and that if the force varies with
distance x as F(x) the problem is handled by slicing
the interval [a,b] into infinitesimal subintervals of length
dx from x to x + dx over which the force can
be considered constant. The displacement from x to x + dx
involves an infinitesimal amount of work dW = F(x)dx, and
these infinitesimal amounts of work are summed by the integral: the
total work
W is the integral of dW from a to b.
Be able to to convert problems of this type into integrals.
[Examples 1 and 2 page 472, Exercises 1-6].
A different application of Calculus is to problems where
the distance that points are moved varies with the parameter x.
In the simplest examples (e.g. emptying a rectangular tank of water
over the top) the force is uniform, but
points at one end of the problem (the bottom) have
to travel farther than those at the other end (the top; here x
is height). The method is to slice the problem perpendicular
to the x-direction, and to let dW be work
corresponding to lifting out the slice at height x and thickness
dx. [Example 3, Exercises 9, 11, 15].
In more complicated examples, the size of the slice may also vary
with x [Example 4, Exercise 17]. Or the
force may also vary with x [Exercise 13].
Be able to apply the "slice and integrate" method to all these
problems. Hydrostatic Pressure and Moments are not covered.
6.7 Probability. Be able to check whether a given function f(x) can be a probability density function [Examples 1,2 page 487, 488] and how to interpret the integral from a to b of f(x)dx as a probability [Exercises 1, 3, 5]. Be able to calculate the average value (= the mean) of the random variable described by a probability density function f(x) [Example 3 page 489, Exercise 6c]. Be able to work with probability density functions given by graphs [Exercise 6] and equations [Example 1]; and in particular with exponentially decreasing probability density functions [Example 2, Examples 3 and 4 pages 489, 490, Exercises 7,9].
7.2 Direction fields. Understand how the
direction field for the first-order differential equation
y' = f(x,y) allows one to "see" the family of
solutions [Figures 1-4 page 505]. Be able to sketch the
direction field of y' = f(x,y) [Example 1, Exercises
9,10, Exercises 11-14 -choose a 5 by 5 grid of points centered on the
initial value]. Given a direction field be able to sketch
solutions, and be able to sketch the solution with a given
initial value [Figures 7,8 page 506; Example 2; Exercises
1a, 3, 7, 18].
Euler's Method. Given a first-order differential
equation y'=f(x,y), an interval [0,T],
an initial value y0 and a number N,
be able to apply Euler's method with N steps to find
approximate values y(T/N), y(2T/N), ... , y(T)
for the solution satisfying y(0) = y0
[Example 3 page 509, Example 4 page 510, Exercises 21-24].
7.3 Separable equations. Understand what a separable equation
is and be able to solve it by integration, applying the
"separate, integrate, solve" method [Examples 1a, 2 page 514].
Understand how the "+C" from
one of your integrals turns into the undetermined constant
in the general solution, and how an initial value determines
what that constant must be [Example 1b, Exercises 9-14].
Mixing problems. Be able to apply the
"rate in - rate out" method to convert a mixing problem to
a separable differential equation, and then be able to solve the
equation [Example 6, problems 35, 37].