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A take-home exam for math 206 section a, focusing on integration and vector calculus. Students are required to sketch regions of integration, evaluate integrals using cartesian, spherical, and cylindrical coordinates, find mass of a sphere, and calculate line integrals. No numerical methods are allowed.
Typology: Exams
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Show all your work to receive full credit for a problem.
Attach this sheet to the solutions you hand in. Even if you attempt the problems in any order, write the solutions in the chronological order.
∫ (^1)
0
√y
2 + x^3 dx dy.
x^2 + y^2 and above by the sphere x^2 + y^2 + z^2 = 1. Write
S f(x, y, z)^ dV^ as an iterated integral in Cartesian coordinates, spherical coordinates and cylindrical coordinates. Do not evaluate any of the three integrals.
4 − x^2 − y^2.
(a) Evaluate
C F~ · d~x.
(b) Evaluate
M curl^ F~ · ~n dσ.
(c) How do the answers in parts (a) and (b) compare? How are C and M related?
