The density function for a femur can be modeled as
ρ(x)=80+πcos(πx)+3x grams per centimeter, −25≤x≤25.
(a) Calculate the total mass of a femur.
(b) Find the center of mass of a femur.
To calculate the total mass of a femur, we need to find the integral of the density function over the range of x values.
(a) To calculate the total mass, we need to integrate the density function ρ(x) with respect to x over the given range:
m = ∫ρ(x) dx
For the given density function ρ(x) = 80 + πcos(πx) + 3x, we integrate it over the range -25 to 25:
m = ∫(80 + πcos(πx) + 3x) dx
To solve this integral, we split it into three separate integrals:
m = ∫80 dx + ∫πcos(πx) dx + ∫3x dx
Integrating each term separately:
m = 80x + (1/π)sin(πx) + (3/2)x^2 + C
Now we evaluate the integral over the given range -25 to 25:
m = (80*25 + (1/π)sin(π*25) + (3/2)*25^2) - (80*(-25) + (1/π)sin(π*(-25)) + (3/2)*(-25)^2)
Simplifying this expression will give us the total mass of the femur.
(b) To find the center of mass of a femur, we need to find the integral of the product of the density function and the position function, divided by the total mass. The position function is simply x.
The center of mass is given by:
x̄ = (1/m) ∫x * ρ(x) dx
We already obtained the value of m in part (a). Now we need to evaluate the integral:
x̄ = (1/m) ∫x * (80 + πcos(πx) + 3x) dx
This integral can also be split into three separate integrals:
x̄ = (1/m) ∫80x dx + (1/m) ∫πxcos(πx) dx + (1/m) ∫3x^2 dx
Integrating each term separately:
x̄ = (1/m) (40x^2 + (1/π^2)sin(πx) - (3/2)x^3) + C
Now we evaluate the integral over the given range -25 to 25:
x̄ = (1/m) ((40*25^2 + (1/π^2)sin(π*25) - (3/2)*25^3) - (40*(-25)^2 + (1/π^2)sin(π*(-25)) - (3/2)*(-25)^3))
Simplifying this expression will give us the center of mass of the femur.