Day 42, Mat129: Calculus I

• Your 4.5 homework is graded: pp. 336, #16, 25, 39
• Your exam revisions are also returned.
• Your 5.1 homework is due.
• Thanks to Felix for alerting me that a new class -- mat128! -- has been added to his Blackboard courses. That is an assessment, which
  • will be available Wednesday;
  • will count as two homework assignment GOHF cards, if you do it (and all I'll know is whether you did it or not);
  • will not affect your grade otherwise. Do your best, that's all I'd ask: take it seriously.

• Monday: final examples of integrals
• Wednesday: review of new material
• Friday: review of all material
• Sunday night, 7:00-9:00p.m.: review, here.
• Next Monday: I'll be in from about 8:00 until exam time.

1. A second type of volume integral
2. A mass problem
3. A work problem

But in each case, we have a different type of quantity that we're after:

1. A volume integral: $V=\int{dV}$
2. A mass problem: $M=\int{dM}$
3. A work problem: $W=\int{dW}$

1. Let's compare the idea of using cross-sectional areas as disks and washers with the volume computed using cylindrical shells (the subject matter of 5.3):
   • Example 4, p. 366 looks a lot like our manipulative
   • I don't like what our book does: Stewart writes
     $V=2\pi rh\Delta r$

     What would I have him write?

     $\Delta V=2\pi rh\Delta r$

     and then

     $V = \int dV \approx \sum_{i=1}^n \Delta V_i= \sum_{i=1}^n 2\pi r_i f(r_i) \Delta r$

   • Let's try #8, p. 367

2. What is the mass of the atmosphere?
   
   • We need to know a little about the Earth (whose radius is 6,360 km).
   • We might propose a linear model for the mass density of the atmosphere, based on the data in the graphic. Let's figure that out....
   • The surface area of a sphere: $4\pi{r^2}$ How can we figure that out?
     • Using cross-sectional area, we discovered that
       $V(r)=\frac{4}{3}\pi r^3$

       so

       $V(r+h)=\frac{4}{3}\pi (r+h)^3$

       The difference in the volumes is essentially the surface area $SA(r)$ times $h$ ($= \Delta r$): hence

       $SA(r)\approx \frac{V(r+h)-V(r)}{h}$

       What would we do to make it exact? This is your final exam....

   • Here's a target: NASA says that "The total mean mass of the atmosphere is $5.1480\times 10^{18}kg$....". Let's see if we get close....

3. Let's now compute the work required to fill a spherical water tank with water.
   • The sphere is of radius 2 meters.
   • The center of the sphere is 20 meters above the surface of the source of water.
   • Water has a density of $1000kg/m^3$
   What is the work required to lift the water into the tower?
   • Work is technically defined as force times displacement: W=F*d
   • In this case, we have the force of gravity lifting layers of water up to the tower: dW=F*dh

   An interesting historical example: #30, p. 373