8.4 (Density and Centre of Mass)

Question 1

What is density in calculus?

Answer 1

Density ρ(x) is the measure of mass per unit length

Question 2

what is the formula to calculate the mass of a one-dimensional object (like a rod) with variable density ρ(x)?

Answer 2

The mass M of the object from x=a to
x=b is:

M=∫a to b ρ(x)dx

Question 3

What is the formula to calculate the moment of a mass distribution along a line?

Answer 3

Mx=∫a to b xρ(x)dx

Question 4

How do you find the center of mass
𝑥ˉ of a one-dimensional object?

Answer 4

xˉ= Mx/M
​
where Mx is the moment and M is the total mass.

Question 5

How do you calculate the mass of a two-dimensional region with density
ρ(x,y)?

Answer 5

M=∬R on the bottomρ(x,y)dA

Question 6

How do you find the center of mass
(𝑥ˉ,𝑦ˉ) of a two-dimensional region?

Answer 6

xˉ=1/M∬ R ρ(x,y)dA

𝑦ˉ=1/𝑀∬𝑅𝑦𝜌(𝑥,𝑦)𝑑𝐴