121616949-math.218

# 121616949-math.218 - compute it ﬁrst M = Z π 2-π 2 cos...

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204 Chapter 9 Applications of Integration Next we do the same thing to find ¯ y . The mass of the plate between y i and y i +1 is approximately n i = y Δ y , so M x = Z 1 0 y y dy = 2 5 and ¯ y = 2 5 3 2 = 3 5 , since the total mass M is the same. The center of mass is shown in figure 9.15 . EXAMPLE 9.21 Find the center of mass of a thin, uniform plate whose shape is the region between y = cos x and the x -axis between x = - π/ 2 and x = π/ 2. It is clear that ¯ x = 0, but for practice let’s compute it anyway. We will need the total mass, so we
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Unformatted text preview: compute it ﬁrst: M = Z π/ 2-π/ 2 cos x dx = sin x ﬂ ﬂ ﬂ π/ 2-π/ 2 = 2 . The moment around the y-axis is M y = Z π/ 2-π/ 2 x cos x dx = cos x + x sin x ﬂ ﬂ ﬂ π/ 2-π/ 2 = 0 and the moment around the x-axis is M x = Z 1 y · 2 arccos y dy = y 2 arccos y-y p 1-y 2 2 + arcsin y 2 ﬂ ﬂ ﬂ ﬂ ﬂ 1 = π 4 . Thus ¯ x = 2 , ¯ y = π 8 ≈ . 393 ....
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