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 rho is a density function, then rhoave 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 = 4 − x2 − y2 and the plane z = 0.
Answers
Step-by-step explanation:
The volume of EE is
\displaystyle V(E)=\iiint_E\mathrm dVV(E)=∭
E
dV
To compute the integral, convert to cylindrical coordinates:
x=r\cos\thetax=rcosθ
y=r\sin\thetay=rsinθ
z=zz=z
\implies\mathrm dV=r\,\mathrm dr\,\mathrm d\theta\,\mathrm dz⟹dV=rdrdθdz
\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}2V(E)=∫
0
2π
∫
0
3
∫
0
9−r
2
rdzdrdθ=
2
81π
Now integrate ff over EE . 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∭
E
3x
2
z+3y
2
zdV=3∫
0
2π
∫
0
3
∫
0
9−r
2
r
3
zdzdrdθ=
8
6561π
Then the average value of ff over EE is \dfrac{\frac{6561\pi}8}{\frac{81\pi}2}=\dfrac{81}4
2
81π
8
6561π
=
4
81
.