MAT 2322 Final: MAT 2322 University of Ottawa Final Exam exam 2+Solutions
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0 x ln x dy dx = z e2. Intersection between the sphere and the plane is given by x2 + y2 + 32 = 25 x2 + y2 = 25 9 = 16. So the region can be described in cylindrical coordinates: ~f (~r(t)) = t2~i + 4t3 ~j + 6t2 ~k, ~r (t) = ~i + ~j + 4t~k, ~f (~r(t)) ~r (t) = t2 + 28t3, ~f d~r = 0. (2,0) to (0,1) ~f (~r(t)) = 2t~i + 7(2 2t)~j, ~f (~r(t)) ~r (t) = 4t + 7(2 2t) = 14 18t, Thus, rc: scalar curl of ~f is . If s is the region bounded by c, then, by green"s theorem, 5 da = 5 (area of s) = 5. 6xyz2)~j + [(2xz3 + 2y) (2xz3 + 2y)]~k. That is, c does not depend on y. So we can write: f (x, y, z) = x2yz3 + xy2 + c(z).
