1. (a) Parametrise C1 and C2. Hint: Use t : t0 ! t0 as limits when parametrising
C2 and explain why cos(t0) =
and sin(t0) =
(b) Calculate I
where v =
(yi + xj).
(c) Use Green`s theorem and your answer from 1(b) to determine the area of R
and then verify that it is less than ab.
2. (a) Give the cartesian equation for the ellipse used to dene C2.
(b) Show that 9 + 4r cos = 5r is the equation of that ellipse when written in
polar coordinates (r; ). Hint: Square both sides rst.
(c) Calculate ZZ
using polar coordinates. Hint: Integrate with respect to r rst and then .
Explain why the limits on the outer integral should be =
3. If T(r) = T0=r3 is the temperature prole in the region R, then use the previous
results to calculate the average temperature in R when T0 = 1000. Verify that the
average temperature is between the minimum and maximum temperatures in R.