UNIVERSITY OF TORONTO
FACULTY OF ARTS AND SCIENCE
Test 4, March 27, 2020
MAT235Y1Y
Examiners: S.Alexakis, F.Alfaisal, B.Choi,
V.Dimitrov, T.Ens, N.Jung
Duration: 24 hours
This test has 6 pages.
Total: 50 marks
All answers must be your own work.
No marks will be given for a completely wrong solution.
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1
1. (10 marks)
a) (3 marks) Evaluate the line integral
∫
C sinx dy, where C is the
part of the curve y = x2 from the point (0, 0) to the point (pi, pi2).
(A) 2pi (B) pi (C) 0 (D) −pi (E) pi2
Write the (capital) letter of the answer in this box. Only the
answer in the box will be marked. )
b) (4 marks) Evaluate the line integral
∮
C F · n ds, where C is the
circle x2+y2 = 4 with counter clockwise orientation, n is the unit
outward normal vector to the curve C and F(x, y) = 〈x33 , 2y
3
3 〉.
(A) 6pi (B) 12pi (C) 0 (D) pi (E) 4
Write the (capital) letter of the answer in this box. Only the
answer in the box will be marked. )
c) (3 marks) Set up the volume of the solid bounded by the surface
x =
√
1− z, the planes x = 0, y = 0, y = 1 and z = 0 as an
iterated integral in the order of dy dx dz.
Answer
2
2. (10 marks) Let F(x, y, z) = 〈yz, xz, xy〉.
(a) (4 marks) Show that curl F(x, y, z) = ~0.
(b) (6 marks) Use a potential function f of F to solve the line integral∫
C F · dr, where C is the curve given by ~r(t) = 〈et, t2 − 1, 2t+ 1〉 from
t = 0 to t = 1.
3
3. (10 marks) Use transformations x+y = u, y = v to evaluate the inte-
gral
∫∫
D
√
4− (x+ y)2 dA, where D is the trapezoidal region bounded
by the lines x+ y = 1, x+ y = 2, y = 0 and x = 0.
4
4. (10 marks) Evaluate the integral
∫∫∫
E
(x2+z) dV , where E is the solid
that lies within the cylinder x2+y2 = 1 and the sphere x2+y2+z2 = 4,
and above the xy-plane.
5
5. (10 marks) Evaluate the integral
∫∫∫
E
√
x2 + y2 + z2 dV , where E is
the solid bounded by the sphere x2 + y2 + z2 = 4 and above the plane
z = 1.
6