MATH 251 – LECTURE 18
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 13.4–6,8
webAssign: 13.4–6, due 3/7 11:55 p.m.
F: Quiz 12.7
Next week: 13.8–10
webAssign: 13.8–10, opens 3/7 12 a.m.
M W: Kevin
F: no lecture
Help Sessions:
M W 5.30–8 p.m. in BLOC 161
Office Hours:
BLOC 641C
M 12:30–2:30 p.m.
W 2–3 p.m.
or by appointment.
Applications
Let D be a region, and let ρ be a density function on D. Then, the total “mass” is
ZZ
ρ(x, y) dx dy.
D
Exercise 1. Electric charge is distributed over the rectangle [1, 2] × [0, 2] with density ρ(x, y) = 2xy + y 2. Find
the total charge.
Applications
Exercise 2. In a class of 10 students, on an exam, points were rewarded as follows: 3 students got 30 points,
1 student got 29 points, 2 students got 27 points, 2 students got 23 points, 1 student got 17 points, and one
student got 5 points. What was the average?
Applications
Let D be a region, and let ρ be a density function on D. Then, the center of mass (x̄, ȳ) is located at
1
hMx, My i ,
hx̄, ȳi =
m
where m is the total mass, and Mx and My are the moments
ZZ
Mx =
x ρ(x, y) dA
Z ZD
My =
y ρ(x, y) dA.
D
Exercise 3. A lamina occupies the part of the disk x2 + y 2 ≤ 49 in the first quadrant. Find the center of mass
of the lamina if the density at any point is proportional to the square of its distance from the origin.
Applications
Exercise 4. A lamina occupies the part of the disk x2 + y 2 ≤ 49 in the first quadrant. Find the center of mass
of the lamina if the density at any point is proportional to the square of its distance from the origin.
Applications
Let D be a region, and let ρ be a density function on D. Then the n:th moment of ρ in x resp. y is
ZZ
ZZ
µ(n)
xn ρ(x, y) dA,
µ(n)
y n ρ(x, y) dA.
x (ρ) =
y (ρ) =
D
(0)
D
(0)
• The total mass is m = µx = µy .
D
E
(1)
(1)
(1)
(1)
1
• The center of mass is m µx , µy . (In the book: µx = Mx and µy = My .)
(2)
(2)
• Ix = Mx and Iy = My are the moments of inertia.
Exercise 5. Find the moments of inertia of the lamina that occupies the region
D = {0 ≤ x ≤ 1, −1 ≤ y ≤ 1} with the density ρ(x, y) = 2xy 2.
Applications
Exercise 5. Find the moments of inertia of the lamina that occupies the region
D = {0 ≤ x ≤ 1, −1 ≤ y ≤ 1} with the density ρ(x, y) = 2xy 2.