# Center of mass and integrating

#### flame_m13

1. The problem statement, all variables and given/known data
A solid "rough" of constant density ($$\delta$$ =1) is bounded below by the surface z=4y^2,above by the plane z=4, and one the ends by the planes x=1 and x=-1. Find the center of mass...

2. Relevant equations
$$_{M}$$xy =$$\int\int\int$$ z $$\delta$$ dV
Mass = $$\int\int\int$$$$\delta$$dV
then to find the center of mass, the z component would be $$_{M}$$xy / M

3. The attempt at a solution
$$_{M}$$xy = $$\int\int\int$$$$^{4}_{4y^2}$$ z $$\delta$$dzdydx
=$$\int\int$$ [z^2/2]$$^{4}_{4y^2}$$dydx
=$$\int$$$$^{1}_{-1}$$$$\int$$$$^{1}_{0}$$(8-8y^4)dydx
=8$$\int^{1}_{-1}$$[y-y^5/5]$$^{1}_{0}$$ dx =8$$\int$$$$^{1}_{-1}$$[4/5]dx
=8[4x/5]$$^{1}_{-1}$$ = 8[4/5+4/5]= 64/5

For some reason, and it isn't obvious to me why, the solution manual says that this answer should be 128/5. i'm off by a factor of 2, but i don't know what i did wrong. I haven't attempted to find the mass yet, but i think once i understand what I did wrong here, that answer shouldn't be hard to find.

#### HallsofIvy

Why are you integrating, with respect to y, from y= 0 to y= 1? The "cylinder" z= 4y2 intersects the plane z= 4 at y= -1 and 1.

#### flame_m13

Why are you integrating, with respect to y, from y= 0 to y= 1? The "cylinder" z= 4y2 intersects the plane z= 4 at y= -1 and 1.
i don't know why that didn't register with me. thanks for pointing that out. my answer is now 128/5, the same as the book's.

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