Let R be the region shown above bounded by the curve C C1 C

Let R be the region shown above bounded by the curve C = C_1 C_2. C_1 is a semicircle with centre at the origin O and radius 9/5. C_2 is part of an ellipse with centre at (4, 0), horizontal semi-axis a = 5 and vertical semi-axis b = 3. 1. (a) Parametrise C_1 and C_2. (b) Calculate_C v middot 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 piab. 2. (a) Give the cartesian equation for the ellipse used to define C_2. (b) Show that 9 + 4r cos theta = 5r is the equation of that ellipse when written in polar coordinates (r, theta). (c) Calculate integral integral_R 1/r^3 dA using polar coordinates. 3. If T(r) = T_0/r^3 is the temperature profile in the region R, then use the previous results to calculate the average temperature in R when T_0 = 1000. Verify that the average temperature is between the minimum and maximum temperatures in R.

Solution

Green’s Theorem: M dx + N dy = Nx My dA. C R

Proof:

i) First we’ll work on a rectangle. Later we’ll use a lot of rectangles to y approximate an arbitrary o region. d ii) We’ll only do M dx ( N dy is similar). C C direct calculation the righ o By t hand side of Green’s Theorem b M d M dA = dy dx. c R y a c y x integral: M(x, d a Inner y)| c = M(x, d) + M(x, c) b M b Outer integral: M( c) R dA = x, y a M(x, d) dx. F o or the LHS we have M dx = M dx + M dx (since dx = 0 along the sides) C bottom top b a = b M(x, c) dx + M(x, d) dx = M(x, c) M(x, d) dx. a b a So, for a rectangle, we have proved Green’s Theorem by showing the two sides are the same. In lecture, Professor Auroux divided R into “vertically simple regions”. This proof instead approximates R by a collection of rectangles which are especially simple both vertically and horizontally. For line integrals, when adding two rectangles with a common edge the common edges are traversed in opposite directions so the sum is just the line integral over the outside boundary. Similarly when adding a lot of rectangles: everything cancels except the outside boundary. This extends Green’s Theorem on a rectangle to Green’s = Theorem on a sum of rectangles. Since any region can be approximated as closely as we want by a sum of rectangles, Green’s Theorem must hold on arbitrary regions.