Writing an expression for the mass of a slice taken out of a shape

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[reef vapor]

Suppose the volume at height h is V(h). Then:
m(h, h + Δh) = ρ*(V(h + Δh) - V(h))
So, first you need to find V(h).

No. We have a paraboloid, not a cylinder.

You can find its volume as a solid of revolution, for example.

Only for Δh -> 0. If Δh is not infinitesimal, it won't be a cylinder.

Well, it's a solid of revolution. You can find it yourself.

Do you know the formula for a solid of revolution?

Suppose we have a curve y = f(x) defined for a < x < b. We revolve this curve around the x-axis, getting a solid of revolution. Then its volume is:
V = π∫(f(x)^2 dx, a, b)

Yes.

No.

We have z = 4(x^2 + y^2). As x^2 + y^2 = r^2:
z = 4r^2
r^2 = z/4
So, our squared function is r(z)^2 = z/4. Thus:
V(h) = ∫((z/4)dz, 0, h)

Well, the antiderivative is that, yes. We need to find the integral from 0 to h, though.

So, what is V(h)?

Well, it's just h^2/8, right?

No, remember: we want the mass of the slice between h and h + Δh.

So:
m(h, h + Δh) = ρ*(V(h + Δh) - V(h))

So, simplify V(h + Δh) - V(h) first.

Well, I mean, it's just that, though.

Mass, of course.

ρ, not p. It's density, as usual.

Yes.

Well, the volume of a slice is just the difference of volumes.

Also, it's "delta", not triangle.

So, to find the volume of a slice between h and h + Δh, we subtract the volume at h from volume at h + Δh.

Well, we don't need to substitute the value of the density yet.
m(h, h + Δh) = ρ((h + Δh)^2/8 - h^2/8) = (1/8)ρ((h + Δh)^2 - h^2)

Now, to simplify the expression in the brackets, try using the difference of squares formula.

What do you mean?

We have a^2 - b^2 = (a - b)(a + b).

Not sure what you mean about the bases.

What do you mean?

[reef vapor]

Well, you can do that, but it's easier to simplify it using the difference of squares formula.

We do have a difference of squares, after all.

No. That would be for h^2 - Δh^2, not (h + Δh)^2 - h^2.

In our case we get (h + Δh)^2 - h^2 = (h + Δh - h)(h + Δh + h) = Δh(2h + Δh).

So:
m(h, h + Δh) = (1/8)ρΔh(2h + Δh)

Think of what?

Well, you just need to practice a bit more, then.

Oh, wait a minute!

[reef vapor]

We forgot π from this point.

So, rather, we get:
m(h, h + Δh) = (π/8)ρΔh(2h + Δh)

[reef vapor]

This only approximately works when ∆h is very small.