1. (a) Parametrise C1 and C2. Hint: Use t : t0 ! t0 as limits when parametrising

C2 and explain why cos(t0) =

4

5

and sin(t0) =

3

5

.

(b) Calculate I

C

v dr

where v =

1

2

(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

R

1

r3dA

using polar coordinates. Hint: Integrate with respect to r rst and then .

Explain why the limits on the outer integral should be =

2

.

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.

2019-01-19T08:44:00+00:00
Maths