center mass of a rod

Question

If you have a rod of length 5 with density p(x)=2+(x/5). What is the center of mass of the rod?

a. ¯x=2.5
b. ¯x=2.667
c. ¯x=3.333
d. ¯x=2.165

Answer 1

Integrate xp(x) from 0 to 5

int (2x+x^2/5) dx  from x=0 to x=5

Antiderivative is x^2 +x^3/15

Evaluate at limits to get x= 33.333

Now integrate p(x) from 0 to 5

int (2 +x/5) dx  from 0 to 5

Antiderivative is 2x +x^2/10

Evaluate at limits to get 12.5

Divide 33.333/12.5 to get x= 2.667 (b)