Let P = (a, b) and let Cr be the circle of radius r cent
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Let P = (a, b) and let Cr be the circle of radius r centered at P. The average value of a continuous function ? on Cr is
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Let P = (a, b) and let Cr be the circle of radius r centered at P. The average value of a continuous function ? on Cr is defined as the integral I?(r) = 1 2? 2? 0 ?(a + r cos ?,b + r sin ?)d? (a) Show that ?? ?n (a + r cos ?,b + r sin ? ) = ?? ?r (a + r cos ?,b + r sin ? ) (b) Use differentiation under the integral sign to prove that d dr I?(r) = 1 2? r Cr ?? ?n ds (c) Use Exercise 41 to conclude that d dr I?(r) = 1 2? r D(r) ? dA where D(r) is the interior of Cr.
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