Question

Let R be the region shown above bounded by the curve C = C1[C2. C1 is a semicircle with center

at the origin O and radius 9

5 . C2 is part of an ellipse with center at (4; 0), horizontal semi-axis

a = 5 and vertical semi-axis b = 3.

1. Let R be the region shown above bounded by the curve C - C1 UC2. C1 is a semicircle with centre at the origin O and radius 음. C2 is part of an ellipse with centre at (4,0), horizontal semi-axis a- 5 and vertical semi-axis b- 3. (a) Parametrise C1 and C2. Hint: Use t: -to - to as limits when parametrising C2 and erplain why cos(to)--^ and sin(to) (b) Calculate v dr (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 παό. 2. Consider again the obove picture of the region R, where C2 is the curve of a part of an ellipse in R. (a) Give the cartesian equation for the ellipse used to define C2 (b) Show that 9+47 cos θ br Is the equation of that ellipse when written in polar coordinates (r,0). Hint: Square both sides first. (c) Calculate dA 3 using polar coordinates. Hint: Integrate with Tespect to T first and then θ. E!plain why the limits on the outer integral should be θ- TO/r3 is the temperature profile in the region R, then use the previous results to calculate If T(r) the average temperature in R when To - 1000. Verify that the average temperature is between the minimum and maximum temperatures in R 3.

Thanks a lot for your help:)

Answer 1

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