High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Calculus 2 Textbook > Bonus: Integral calculus in several variables (Videos Coming Soon)

Applications of multiple integrals

Improper IntegralsPrevious Section
Practice quiz: Multiple integralsNext Section
Popular Courses
Find My Course

Moments and centre of mass

Concept (Moments and Centres of Mass). Let δ=δ(x,y)\delta = \delta (x, y) be the density of a region R in the xy-plane. Its mass is given by
m=∬RδdAm =\iint\limits_R\delta dA
The moment of the region about the y-axis and x-axis are given respectively by
Mx=0=∬RxδdAMy=0=∬RyδdAM_{x=0} =\iint\limits_Rx\delta dA\hspace*{70pt} M_{y=0} = \iint\limits_Ry\delta dA
The x and y-coordinates for the centre of mass are given respectively by
x‾=Mx=0m=∬RxδdA∬RδdAy‾=My=0m=∬RyδdA∬RδdA\overline{x} = \frac{M_{x=0}}{m} =\frac{\iint\limits_Rx\delta dA}{\iint\limits_R\delta dA}\hspace*{70pt} \overline{y} = \frac{M_{y=0}}{m} =\frac{\iint\limits_Ry\delta dA}{\iint\limits_R\delta dA}
In 3 dimensions, if δ=δ(x,y,z)\delta = \delta(x, y, z) is the density of a region R in the xyz-space, then its mass is given by
m=∭RδdVm = \iiint\limits_R\delta dV
The moment of the region about the yz-plane, xz-plane, and xy-plane are given respectively by
Mx=0=∭RxδdVMy=0=∭RyδdVMz=0=∭RzδdVM_{x=0} = \iiint\limits_Rx\delta dV\qquad M_{y=0} = \iiint\limits_Ry\delta dV \qquad M_{z=0} = \iiint\limits_Rz\delta dV
The x, y, and z-coordinates for the centre of mass are given respectively by
x‾=Mx=0m=∭RxδdV∭RδdVy‾=My=0m=∭RyδdV∭RδdVz‾=Mz=0m=∭RzδdV∭RδdV\overline{x} = \frac{M_{x=0}}{m} = \frac{\iiint\limits_R x\delta dV}{\iiint\limits_R \delta dV}\qquad \overline{y} = \frac{M_{y=0}}{m} = \frac{\iiint\limits_R y\delta dV}{\iiint\limits_R \delta dV}\qquad \overline{z} = \frac{M_{z=0}}{m} = \frac{\iiint\limits_R z\delta dV}{\iiint\limits_R \delta dV}
→\rightarrow Notes:
  1. If δ\delta is a constant, then it factors out of the expressions for centre of mass, and centre of mass is called the centroid or centre of gravity of the region.
  2. You can use geometric arguments to find some coordinates of the centroid. For example, if the object is rotationally symmetric about the z-axis, then x‾=y‾=0.\overline{x} = \overline{y} = 0.

Practice: Centroid

Find the centroid of the region bounded by z = x2 + y2 and z = 4

Moment of inertia

Concept (Moment of Inertia). Let δ=δ(x,y,z)\delta = \delta (x, y, z) be the density of a region R in the xyz-space. The moment of inertia of the region about the x, y, and z-axes are given by
Ix=∭R(y2+z2)δ dVIy=∭R(x2+z2)δ dVIz=∭R(x2+y2)δ dVI_x = \iiint\limits_R(y^2+z^2)\delta\, dV\qquad I_y = \iiint\limits_R(x^2+z^2)\delta \,dV\qquad I_z = \iiint\limits_R (x^2 + y^2)\delta\, dV
More generally, the moment of inertia of the region about an arbitrary axis is given by
I=∭RD2δ dVI = \iiint\limits_RD^2\delta\, dV
where D is the perpendicular distance to the axis.

Practice: Moment of inertia

Consider the cylinder of constant density δ\delta defined by x2 + y2 = 4, bounded between z=−1z = -1 and z = 1. Find its moment of inertia when rotated about the z-axis.