Question

The average value of a function f(x, y, z) over a solid region E is defined to be fave = 1 V(E) E f(x, y, z) dV where V(E) is the volume of E. For instance, if ρ is a density function, then ρave is the average density of E. Find the average value of the function f(x, y, z) = 3x2z + 3y2z over the region enclosed by the paraboloid z = 9 − x2 − y2 and the plane z = 0.

Answer 1

The volume of $E$ is

$\displaystyle V(E)=\iiint_E\mathrm dV$

To compute the integral, convert to cylindrical coordinates:

$x=r\cos\theta$

$y=r\sin\theta$

$z=z$

$\implies\mathrm dV=r\,\mathrm dr\,\mathrm d\theta\,\mathrm dz$

$\displaystyle V(E)=\int_0^{2\pi}\int_0^3\int_0^{9-r^2}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta=\frac{81\pi}2$

Now integrate $f$ over $E$. In cylindrical coordinates, we get

$\displaystyle\iiint_E3x^2z+3y^2z\,\mathrm dV=3\int_0^{2\pi}\int_0^3\int_0^{9-r^2}r^3z\,\mathrm dz\,\mathrm dr\,\mathrm d\theta=\frac{6561\pi}8$

Then the average value of $f$ over $E$ is $\dfrac{\frac{6561\pi}8}{\frac{81\pi}2}=\dfrac{81}4$.