Michael Corral: Vector Calculus

124 CHAPTER 3. MULTIPLE INTEGRALS
3.6 Application: Center of Mass
Recall from single-variable calculus that for a regionR={(x,y)
: a≤ x≤b,0≤y≤ f(x)}in 2 thatrepresents
a thin, flat plate (see Figure 3.6.1), where
f(x) is a continuous function on [a,b], the center of
mass of R has coordinates (¯x,ȳ) given by
where
¯x= M y
M and ȳ= M x
M ,
y
y= f(x)
R
(¯x,ȳ) x
0 a b
Figure 3.6.1 Center of mass of R
M x =
∫ b
a
(f(x)) 2
dx, M y =
2
∫ b
a
xf(x)dx, M=
∫ b
a
f(x)dx, (3.27)
assuming that R has uniform density, i.e the mass of R is uniformly distributed over
the region. In this case the area M of the region is considered the mass of R (the
density is constant, and taken as 1 for simplicity).
In the general case where the density of a region (or lamina) R is a continuous
functionδ = δ(x,y) of the coordinates (x,y) of points inside R (where R can be any
region in 2 ) the coordinates (¯x,ȳ) of the center of mass of R are given by
¯x= M y
M and ȳ= M x
M , (3.28)
where
M y =
xδ(x,y)dA, M x =
yδ(x,y)dA, M=
δ(x,y)dA, (3.29)
R
R
R
The quantities M x and M y are called the moments (or first moments) of the region R
about the x-axis and y-axis, respectively. The quantity M is the mass of the region R.
To see this, thinkof taking asmall rectangleinsideRwith dimensions∆x and∆y close
to 0. The mass of that rectangle is approximatelyδ(x ∗ ,y ∗ )∆x∆y, for some point (x ∗ ,y ∗ )
in that rectangle. Then the mass of R is the limit of the sums of the masses of all such
rectangles inside R as the diagonals of the rectangles approach 0, which is the double
integral δ(x,y)dA.
R
Notethattheformulasin(3.27)representaspecialcasewhenδ(x,y)=1throughout
R in the formulas in (3.29).
Example 3.13. Find the center of mass of the region R={(x,y) : 0≤ x≤1, 0≤y≤2x 2 },
if the density function at (x,y) isδ(x,y)= x+y.