Im currently studying, calculus in spherical method.
I encounter some confusion with the function $f(ρ,θ,φ)$ in the following formula
$$ \displaystyle\iiint_{E} f(ρ,θ,φ) ρ^2\sinφdφdρdθ $$
Taking the volume of say bounded by Hemisphere $x^2 +y^2+(z-3)^2 = 9$ and a cone $z=\sqrt{x^2+y^2}$ some of the previous equations suggests $f(ρ,θ,φ)=1$. I'm a bit confuse if its because previous example takes a unitary distance between origin. Or is there some way i could find the value for $f(ρ,θ,φ)$ in the integrand?
Your question has nothing to do with spherical coordinates. If $A$ is a region of the space $\Bbb R^3$, then$$\operatorname{vol}(A)=\iiint_A1\,\mathrm dx\,\mathrm dy\,\mathrm dz.$$For instance, if $A=[a,b]\times[c,d]\times[e,f]$, then the volume of $A$ is $(b-a)(d-c)(f-e)$, and\begin{align}\iiint_A1\,\mathrm dx\,\mathrm dy\,\mathrm dz&=\int_a^b\int_c^d\int_e^f1\,\mathrm dz\,\mathrm dy\,\mathrm dx\\&=\int_a^b\int_c^d(f-e)\,\mathrm dy\,\mathrm dz\\&=\int_a^b(d-c)(f-e)\,\mathrm dx\\&=(b-a)(d-c)(f-e).\end{align}