MATH200 April 2012
• Q1 (a) • Q1 (b) • Q2 (a) • Q2 (b) • Q3 • Q4 (a) • Q4 (b) • Q4 (c) • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) • Q7 • Q8 • Q9 • Q10 •
The average distance of a point in the plane region D to a point (a,b) is defined by
where A(D) is the area of the plane region D. Let D be the unit disk
Find the average distance of a point in D to the centre of D.
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Without doing any calculations, where is the centre of the unit disc, ? What is the discs' area?
Try making a change of coordinates to simplify the evaluation of the integrals that will show up. Don't forget the Jacobian.
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Since D is the unit disc, we know that it is a circle of radius 1 and so it has area π, i.e.
The centre of D is the origin and so (a,b) = (0,0). To evaluate the integral,
we use polar coordinates:
where we have used the fact that (i.e. the Jacobian of the transformation from Cartesian to Polar coordinates) and . Evaluating the integral gives:
Hence, the average distance from any point within D and the centre of D (the origin) is
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