1. Test Remarks, and Physics (3/2/05) Concerning Test #1: – You may correct your mistakes (by Monday). You will get back 1/3 of the points you lost if your corrections are right. Do this on your own or talk to me. Learn from your mistakes!! – Always look for algebraic simplification first. – Don’t just immediately start using parts method. Plain substitution, if possible, is much simpler. – Improper integrals never automatically converge or diverge. It depends on the function! To find out, either compute the integral or compare it with a known one!

2. A Physics Application: The Mass of Objects with Variable Density In general, the mass of an object is its density times its size (i.e., its length if it’s 1-dimensional, its area if it’s 2-dimensional, and its volume if it’s 3-dimensional). If the density (i.e., mass per unit length, or unit area, or unit volume) of an object varies, however, then we can compute total mass by slicing the object up into parts with (near) constant density, and then adding up.

3. A 1-dimensional example Suppose the density of spectators along the 26 mile Boston Marathon route from Hopkinton to Boston is approximately f(x) = 5000  x people/mile (where x is the distance from Hopkinton). About how many spectators are along the route in total (i.e., what is the “total mass of spectators”)? Solution:

4. A 3-Dimensional Example The density of air decreases as you move away from the earth’s surface (well, duh). The density function is approximately  (h) = 1.28 e -0.000124h kg per cubic meter, where h is the distance from the surface in meters. What is the total mass (in kg) of a cylinder of air 10 meters in diameter and 10 kilometers (i.e. about 6.2 miles) high?

5. Example solved The mass of a cross-sectional slice will be the density there times the volume, which is the constant 25   h. Multiplying by the density and adding them all up, we get

6. Assignment for Friday Correct your test! There will be time for questions on the test Friday. Due Monday. Read the first 3 pages of Section 8.3. On pages 366-7 do #1, 3, 5, 9, 13 Extra Credit Problem (10 points): Prove that the surface area of a sphere of radius r is 4  r 2. Hint: The cross-sections here are circular “bands” whose width is  s (arc length), not  x. (Work on your own – honor code!)