Answer:
The mass of the solid is 16 units.
The center of mass of the solid lies at (0.6875, 0.3542, 2.021)
Work:
Density function: ρ(x, y, z) = 8
x-bounds: [0, 1], y-bounds: [0, x], z-bounds: [0, x+y+3]
The mass M of the solid is given by:
M = ∫∫∫ρ(dV) = ∫∫∫ρ(dx)(dy)(dz) = ∫∫∫8(dx)(dy)(dz)
First integrate with respect to z:
∫∫8z(dx)(dy), evaluate z from 0 to x+y+3
= ∫∫[8x+8y+24](dx)(dy)
Then integrate with respect to y:
∫[8xy+4y²+24y]dx, evaluate y from 0 to x
= ∫[8x²+4x²+24x]dx
Finally integrate with respect to x:
[8x³/3+4x³/3+12x²], evaluate x from 0 to 1
= 8/3+4/3+12
= 16
The mass of the solid is 16 units.
Now we have to find the center of mass of the solid which requires calculating the center of mass in the x, y, and z dimensions.
The z-coordinate of the center of mass Z is given by:
Z = (1/M)∫∫∫ρz(dV) = (1/16)∫∫∫8z(dx)(dy)(dz)
Calculate the integral then divide the result by 16.
First integrate with respect to z:
∫∫4z²(dx)(dy), evaluate z from 0 to x+y+3
= ∫∫[4(x+y+3)²](dx)(dy)
= ∫∫[4x²+24x+8xy+4y²+24y+36](dx)(dy)
Then integrate with respect to y:
∫[4x²y+24xy+4xy²+4y³/3+12y²+36y]dx, evaluate y from 0 to x
= ∫[28x³/3+36x²+36x]dx
Finally integrate with respect to x:
[7x⁴/3+12x³+18x²], evaluate x from 0 to 1
= 7/3+12+18
Z = (7/3+12+18)/16 = 2.021
The y-coordinate of the center of mass Y is given by:
Y = (1/M)∫∫∫ρy(dV) = (1/16)∫∫∫8y(dx)(dy)(dz)
Calculate the integral then divide the result by 16.
First integrate with respect to z:
∫∫8yz(dx)(dy), evaluate z from 0 to x+y+3
= ∫∫[8xy+8y²+24y](dx)(dy)
Then integrate with respect to y:
∫[4xy²+8y³/3+12y²]dx, evaluate y from 0 to x
= ∫[20x³/3+12x²]dx
Finally integrate with respect to x:
[5x⁴/3+4x³], evaluate x from 0 to 1
= 5/3+4
Y = (5/3+4)/16 = 0.3542
The x-coordinate of the center of mass X is given by:
X = (1/M)∫∫∫ρx(dV) = (1/16)∫∫∫8x(dx)(dy)(dz)
Calculate the integral then divide the result by 16.
First integrate with respect to z:
∫∫8xz(dx)(dy), evaluate z from 0 to x+y+3
= ∫∫[8x²+8xy+24x](dx)(dy)
Then integrate with respect to y:
∫[8x²y+4xy²+24xy]dx, evaluate y from 0 to x
= ∫[12x³+24x²]dx
Finally integrate with respect to x:
[3x⁴+8x³], evaluate x from 0 to 1
= 3+8 = 11
X = 11/16 = 0.6875
The center of mass of the solid lies at (0.6875, 0.3542, 2.021)
