Math 254 , Winter 2008
Instructor:
Rodica D. Costin
Office: Math Tower 436
Office
hours: Monday and Friday, 9:30-10:10, and by
appointment.
Tutoring offered at MSLC.
Please check the Final examination schedule.
W Feb. 6 Topics covered: Section 15.3
F Feb. 8 Topics covered: Using symmetries in double integrals and Section 15.4.
M Feb. 11 Topics covered: from 15.5 - density and mass, charge density, moments, moments of inertia, center of mass.
Homework write-ups due in recitation on Tuesday Feb. 12: HW5
Sec 15.1 # 13
Sec 15.2 #6,16,26
Sec 15.3 #12,40
W Feb. 13 Topics covered: Section 15.6, and the set-up of 15.7.
Homework write-ups HW 6 due next Tuesday
sec 15.4 #13,17
sec 15.5 #5,23
sec 15.6 #2,6
Remember: you need to solve many more problems to master the material!
F Feb. 15 Sect. 15.7. Please solve by Monday the problems 8,10,14,16,36.
M Feb. 18 Sect. 15.8 Please solve by Wednesday: 15.8: 10, 18, 20, and set up an iterated integral in spherical coordinated for the region in problem 6.
W Feb. 20 Review
Special office hours on W Feb 20: 9:35-10:30 and 11:35-12:00.
F Feb. 22 Second midterm test
Homework write-ups HW 7 due next Tuesday
Sec 15.7 #4, 12, 20
Sec 15.8 #9, 20, 36
Remember: you need to solve many more problems to master the material!
M Feb. 25 Sec. 15.9
W Feb. 27 More examples from 15.9 and start 16.1.
Homework write-ups HW 8 due next Tuesday
15.9#4,14
16.1#4,24
16.2#10,20
F Feb. 29 Sec. 16.1 and 16.2 Please solve by Monday (for practice) the problems of 16.2: 3,4,11,19.
M March 3 16.3 By Wednesday: please Look at Examples 3, 4, 5 and solve problems: 3,5 (in 16.3).
W March 5 16.4 Please solve the following problems: 2, 3, 4, 7, 8, 15,18.
Green's Theorem: an intuitive picture from fluid mechanics
F March 7 Review. Study guide
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Would you like some more interesting problems? Try these, for bonus credit (extra 3 points each).
Turn them in in lecture (before the last lecture of the quarter).
1. Give a sound argument for Cavalieri's Principle: if two objects of equal heights have all their respective cross-sections of equal area, then the volumes of the objects are equal. Try it: take a stack of coins, identical or not. Now push the coins slightly to change the shape of the stack. (See a picture on Wikipedia.)
2. A (general) cylinder is, by definition, the following surface. Take a curve C (like a circle, or a parabola, or a square etc.) in the xy-plane. Fix a direction in space. Then through all the points of C draw a line in the fixed direction.
For example, if C is a circle, you get a circular cylinder, and if the direction is k, then you get (the familiar) right circular cylinder. Or, if C is a rectangle, then you get a parallelipiped.
Now at height z=h cut your surface by a horizontal plane. Show that the volume enclosed by the surface (assuming C is a closed curve) equals the area enclosed by C times h.
3. A general cone is defined as follows. Consider a curve C in the xy plane and a point V outside the plane. Now draw lines joining V to all the points on C. For example, if C ia a polygon, you obtain a pyramid, and if C is a circle you obtain a circular cone.
Give a sound argument that the volume enclosed by this surface and by the xy-plane equals the area enclosed by C times the distance from V to the xy-plane, divided by 3.