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1. Let F(x, y, z) = (y + 2xyz3, x + x2z3,1 + 3x2yz2).
(a) Show that r ⇥ F = 0.
(b) Find a function f : R3 ! R such that F = r f .
(c) Hence, or otherwise, find the value of the integralπ
C
F · d s,
where C is the part of the curve with parametrisation (t) = (2 cos t,3 sin t,4t/⇡) with
t 2 [0,⇡].
2. Let C be the part of the graph of y =
p
x2 1 between x = 2 and x = 3.
(a) Write down a parametrisation of C, and find the velocity vector of your parametrisation.
(b) Find the line integral π
C
4xy ds.
3. (a) You are given the iterated integralπ 1
0
π px
x3
f (x, y) dydx.
Sketch the domain of integration of this iterated integral (your sketch should be drawn
by yourself not by some software) and rewrite it as an iterated integral in the reverse
order (that is, with inner integral with respect to x and outer integral with respect to y).
(b) Let P be the parallelogram in R2 with vertices (0,0), (1,2), (5,2), and (6,0). Calculate
the double integral ∫
P
(y 2x) dA.
4. Calculate the area enclosed by the curve
(t) = (6 cos t cos(6t),6 sin t sin(6t)), t : 0! 2⇡.
A sketch of the curve is shown below.
-6 -4 -2 2 4 6 x
-6
-4
-2
2
4
6
y
2