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Millersville University Name Department of Mathematics MATH 311, Calculus III , Test 4 December 2, 2005, 8:00A-8:50A
Please answer the following questions. Answers without justifying work will receive no credit. Partial credit will be given as appropriate, do not leave any problem blank.
- (5 points) For an arbitrary continuous function f (x, y) write down the equivalent iter- ated integral in the opposite order of integration to the one shown below. ∫ (^2)
0
x^2
f (x, y) dy dx
- (10 points) For an arbitrary continuous function f (x, y, z) convert the following triple integral to spherical coordinates. ∫ (^1)
− 1
∫ √ 1 −x 2
0
∫ √ 2 −x (^2) −y 2 √ x^2 +y^2
f (x, y, z) dz dy dx
- (10 points) Determine if the following vector field is conservative. If it is, find a potential function. F(x, y) = 〈y cos x, sin x − y〉
- (12 points) Find the volume of the region above z = x^2 + y^2 and below z = 8 − x^2 − y^2.
- (10 points) If F(x, y) = (4xy − 2 x)i + (2x^2 − x)j find the work done moving along y = x^2 from (− 2 , 4) to (2, 4).
- (10 points) Compute the volume of the solid region bounded by the following surfaces.
z = 1 − x^2 , z = 0, y = 2, y = 4
- (13 points) Use Green’s Theorem to evaluate the following line integral. ∮
C
ye^2 x^ dx + x^2 y^2 dy,
where C is the rectangle from (− 2 , 0) to (3, 0) to (3, 2) to (− 2 , 2) to (− 2 , 0).
- (10 points) The gravitational vector field is
F(x, y, z) =
−GmM (x^2 + y^2 + z^2 )^3 /^2
〈x, y, z〉,
where G is the universal gravitational constant, and M and m are (constant) masses. Find the potential energy of the object of mass m at position (a, b, c).
