Math 104 - Lecture 4 Notes Thursday, September 29, 2011

Week 4 Material: Chapter 8 - Methods of integration,

• Trigonometric integrals and trig substitutions

• Chapter 8 -- sections 8.2 and 8.3

Maple - int

Fourth Homework Assignment - due at lecture on Thursday, October 13.

1. Reading: Sections 8.2 and 8.3
2. More reading: In the Maple/Calculus Lab Manual, read the section of the manual on "int".
3. Textbook assignment: (submit nothing) Make certain you can do the following problems from the textbook:
• Section 8.2, page 466, #5, 15, 19, 23, 31, 37, 69
• Section 8.3, page 470, #3, 7, 15, 25, 35, 43, 55
4. Textbook assignment: Submit answers to the following textbook problems in lecture on Thursday, October 13:
• Section 8.2, page 466, #8, 18, 22, 28, 44, 64, 72
• Section 8.3, page 470, #4, 8, 20, 32, 36, 52, 58

BONUS PROBLEM!!

Bonus problem #4: A cylindrical glass of radius r and height L is filled with water and then tilted until the water remaining in the glass exactly covers its base.

• (a) Determine a way to "slice" the water into parallel rectangular cross sections and then set up a definite integral for the volume of the water in the glass.
• (b) Determine a way to "slice" the water into parallel cross-sections that are trapezoids and then set up a definite integral for the volume of the water.
• (c) Find the volume of water in the glass by evaluating one of the integrals in part (a) or part (b).
• (d) Find the volume of water in the glass from purely geometric considerations.
• (e) Suppose the glass is tilted until the water covers half the base. In what direction can you "slice" the water into triangular cross sections? Rectangular cross-sections? Cross-sections that are segments of circles? Find the volume of water in the glass.
• Here are two pictures that may help:
  

For full credit, you must be as specific as you possibly can! When you have solved it, either write up the solution by hand and give it to me, or else send the solution to

deturck@math.upenn.edu

by electronic mail - extra credit for the first complete solution I receive.