MATH 201 Problems Extra Problems for Part III, Fall 2012
American University of Beirut · Calculus and Analytical Geometry III · 7 pages · Prof. Nazih Nahlus
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Math 201 Extra Problems for Part III
N. Nahlus
Final Exam.
Parts I, II: 22% on chapter 10, 22% on chapter 14
Part III: 38% on chapter 15, 18% on chapter 16.
1. Find the value of (Change the order of integration) (Fubini’s Thm.)
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2. Evaluate (Hint: Change to polar. )
3. Find the area of the region that is inside the lemniscate & inside the circle .
Hint:
4a. Set up (but do not evaluate) the triple integral(s) in Cylinderical
coordinates that represent the volume
of the region in the 1st octant bounded by the cylinder and the surface
4b. Set up (but do not evaluate) the triple integral(s) in Cartesian coordinates that represent the volume of the
region in the first octant bounded by the coordinate planes below the surface
5. a) Set up (but do not evaluate
) the triple integral(s) in spherical coordinates in the order d d d
that represent the volume of the region inside the sphere and inside
b) Set up (but do not evaluate) the triple integral(s) for the above volume in the order d d d.
Note: This is problem 31 in section 15.7 whose answer appears at the end of Thomas book.
Note: All curves C below are assumed to be smooth (except for finitely many pints)
counter clockwise, and traced once.
6a. Use Green’s Theorem to find the circulation and the (outward) Flux
and C is the circle:
6b. Evaluate for any path C (not passing through the origin) from A(1,1) to B(3,2) by finding a
potential function f(x,y) by inspection.
6c. How would you solve 14b without finding a potential function f(x,y)?
(For simplicity, assume C is in the simply connected region R= The plane minus the negative x-axis).
(Hint ++: Use the 3 theorems in 16.3 to replace C by a straight line from A to B.)
7a. Use Green’s Theorem (in a clever way) to find
where C is the circle . (Hint ++: Use the half-substitution trick).
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7b. Use Green’s theorem (with holes) to show that for any simple closed curve C around the
origin. 7c
: In 7b, what happens if C is not surrounding the origin.
Hint: Check except at the origin.
7d) What are the ----- 3 secrets----- in proving Green’s Theorem (Circulation version)
8) (15.8) Thomas Problems in 15.8: 22, 23, 9
9) 16.4: Lecture Problems Level **** (like Green’s Theorem for non-closed curves, …etc)
10) 16.4: Thomas : 17 (Flux), 27(Area) (just set up integrals)
Thomas: 37 (very easy), 39 (where F=grad(f) where f=…...) Hint: For 39a, see 37.
11) More problems on 15.2 and 15.4 (like problems 1, 2, 3 above & others)
(Start with Lecture problems -----then check Thomas book)
12) More problems on 15.5, 15.7 (Start with Lecture problems----- then check Thomas book)
13) 15.6: Formulas for: center of mass & Iz (the 2nd moment of inertia around z-axis)
Good Luck