Section 8.4 Worksheet
1. Find the mass of:
a.) a 3 X 4 box with a density of 𝛿 = 2ℎ gm/cm
b.) a rod located at −1 and 1 with a density of 𝛿 = 𝑥 2 gm/m
c.) a rod located with ends at ±1 = 𝑥 and a density of 𝛿 = 3 − 𝑒 −𝑥
2. Find the center of mass of:
a.) an area bounded by 𝑥 = 0, 𝑥 = 1, 𝑥-axis, and 𝑦 = √𝑥 with a density of 𝛿 = 5 k/𝑚2
b.) a system with a mass of 2𝑔 located 3𝑚 to the left of the origin, a mass 5𝑔 located
4𝑚 to the right of the origin, and a mass of 6𝑔 located 6𝑚 to the right of the origin.
3. A plate occupying the region 0 ≤ 𝑥 ≤ 2, 0 ≤ 𝑦 ≤ 3 has a density of 𝛿 = 5 g/𝑚2 . Set up two
integrals giving the mass of the plate, one corresponding to strips in the 𝑥-direction and one
corresponding to strips in the 𝑦-direction. Then calculate both.
4. Find the center of mass of a 4 meter rod with a density of 𝛿 = 2 + 6𝑥 g/𝑚2
5. The density of cars on the Pennsylvania Turnpike, measured in terms of 𝑥 miles to the
Willow Grove Toll, is given by 𝛿(𝑥) = 300(2 + sin(4𝑥 + 0.15)). Find the total number of cars
on the 20 mile stretch.