ASSIGNMENT 9
Problem 1 (7 marks) Evaluate the integral
∫∫
D
x−y+2
x+y+1
dA, where D is the the region
bounded by the lines x + y = 2, x− y = −2, x = −y, and x = y.
Problem 2 (13 marks) Let C be the intersection of the sphere x2 + y2 + z2 = 1 and the
plane y + z = 1. C is oriented counterclockwise (when viewed from above).
(a) Evaluate
∮
C
z dx + x dy + y dz.
(b) Evaluate
∮
C
x2z ds.
Problem 3 (3 marks) Let F(x, y, z) = 〈y, x+2y sin z, z+y2 cos z〉 be a conservative vector
field. Evaluate the line integral
∫
C
F · dr, where C is a curve from the point (1, 2, 0) to the
point (−1, 1, pi
2
). (See the tutorial 16 question).
Problem 4 (7 marks) Use Green’s theorem to evaluate the line integral
∮
C
(y+x tan−1 x) dx+
(xy) dy, where C is the boundary of the region enclosed by the curves y = x and y =
√
x,
oriented counterclockwise.
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