SOLUTIONS TO PRACTICE PROBLEMS FOR THE FINAL
EXAM
1. (a) (e−1)/2. (b) (1.136,1.652). 2. 1/2.
3. (a) 3π/2. (b) π
8ln 5. 4. 1/12; (2/5,1/5,2/5).
5.
(a)Z√2
−√2Z√2−x2
−√2−x2Z√4−x2
−y2
√x2+y2
(x+ 2y+z)dzdydx,
(b)Z2π
0Z√2
0Z√4−r2
r
r(rcos θ+ 2rsin θ+z)dzdrdθ,
(c)Z2π
0Zπ/4
0Z2
0
ρ3(sin ϕcos θ+ 2 sin ϕsin θ+ cos ϕ) sin ϕ dρdϕdθ.
6. (a) (a) 2ma3/3. (b) π/14.
7.
Zπ/2
0Z2
1Z√4−r2
0
r(z2+r)dzdrdθ.
8.
Z2π
0Z2
0Z√16−r2+6rsin θ
−√16−r2+6rsin θ
r dzdrdθ
9. (i) π−9/2. (ii) −3π/4. The force field is not conservative because
Rc~
F·d~r is not path independnt.
10. 4/5.
11. (a) 2π,~
Fis not conservative. (b) curl ~
F= 0. This does not contradict
Green’s Theorem because the partial derivatives of ~
Fare not continuous at
the origin.
12. (a) 10. (b) 17.
13. No, since div(curl ~
G) must be 0 but div(2x~
i+ 3yz ~
j−xz2~
k) = 2 + 3z−
2xz 6= 0.
14. (a) (2a/π, 2a/π). (b) (3a/8,3a/8,3a/8). 15. 500,000π.
16. 11π.17. −1/2. 18. 4π.
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