Find the centroid of the solid in part (a).

[Litcyb]

Homework Statement

Find the volume of the solid that lies above the cone
ϕ=pi/3
and below the sphere
ρ=4cosϕ
.

Homework Equations

Find the centroid of the solid in part (a).


.

The Attempt at a Solution

For the volume I got 10pi which I am fairly sure is correct. I attempted trying to calculate the centroid and i got a different answer from the book. I have (0,0,21/40)

and I would like to know what did i do incorrect. I calculated ybar and xbar and i got 0.
for zbar, i did the following

∫0→pi/3∫0→2pi∫0→4cosϕ (ρ^3cosϕsinϕdρdθdϕ)


the book's answer for the inner integration of rho was ∫0→2pi∫0→pi/3 (cosϕsinϕ(64cos^4(ϕ))dϕdθ

is this correct? isn't supposed to be ∫0→2pi∫0→pi/3 (cosϕsinϕ(64cos^4(ϕ)/4)dϕdθ?

or am I wrong? (I divided by 4, because that's the rule of integration)

 

[Dick]

Homework Statement

Find the volume of the solid that lies above the cone
ϕ=pi/3
and below the sphere
ρ=4cosϕ
.

Homework Equations

Find the centroid of the solid in part (a).


.

The Attempt at a Solution

For the volume I got 10pi which I am fairly sure is correct. I attempted trying to calculate the centroid and i got a different answer from the book. I have (0,0,21/40)

and I would like to know what did i do incorrect. I calculated ybar and xbar and i got 0.
for zbar, i did the following

∫0→pi/3∫0→2pi∫0→4cosϕ (ρ^3cosϕsinϕdρdθdϕ)


the book's answer for the inner integration of rho was ∫0→2pi∫0→pi/3 (cosϕsinϕ(64cos^4(ϕ))dϕdθ

is this correct? isn't supposed to be ∫0→2pi∫0→pi/3 (cosϕsinϕ(64cos^4(ϕ)/4)dϕdθ?

or am I wrong? (I divided by 4, because that's the rule of integration)

You get (4cosϕ)^4/4, right? What's 4^4/4?

 

[Litcyb]

hahaha!I fell so dumb now. Making very stupid mistakes. thanks for output!