Use the result of Exercise 25 to compute the moments of the semicircle x 2 + y

Question:

Use the result of Exercise 25 to compute the moments of the semicircle x2+y2=R2,y0 as line integrals. Verify that the centroid is (0,4R/(3π)).


Data From Exercise 25

The Centroid via Boundary Measurements The centroid (see Section 15.5) of a domain \(\mathcal{D}\) enclosed by a simple closed curve \(C\) is the point with coordinates \((\bar{x}, \bar{y})=\left(M_{y} / M, M_{x} / Might)\), where \(M\) is the area of \(\mathcal{D}\) and the moments are defined by

\[
M_{x}=\iint_{\mathcal{D}} y d A, \quad M_{y}=\iint_{\mathcal{D}} x d A
\]

Show that \(M_{x}=\oint_{C} x y d y\). Find a similar expression for \(M_{y}\).


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Calculus

ISBN: 9781319055844

4th Edition

Authors: Jon Rogawski, Colin Adams, Robert Franzosa

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