Question

http://www.chegg.com/homework-help/questions-and-answers/find-average-value-function-f-x-y-z-x2z-y2z-region-enclosed-theparaboloid-z-1-x2-y2-plane--q144398

Answer 1

@phi

Answer 2

The average value is the integral of f(x,y,z) dV over the volume, divided by the total volume of the region

\[ \overline{f} = \frac{1}{V}\int \int \int _V f \ dV \]

The first step is to find the volume of the given region, a paraboloid
I would use cylindrical coordinates.

Answer 3

so @phi \[\int\limits_{-1}^{1}\int\limits_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\int\limits_{0}^{1-x^2-y^2} x^2z+y^2z \] dz dy dx

Answer 4

do you know about cylindrical coordinates? r, theta, and z ?

Answer 5

I would use
x= r cos A
y = r sin A
and (x^2 +y^2) z = r^2 z
\[ \frac{1}{V}\int_0^1 \int_0^{2\pi} \int_0^{\sqrt{1-z}} r^2\ z\ r\ dr\ dA \ dz \]
where dV= r dr dA dz replaces dx dy dz
and
\[ V= \int_0^1 \int_0^{2\pi} \int_0^{\sqrt{1-z}} r\ dr\ dA \ dz \]

Answer 6

how do we evaluate this ? stuck at dA

Answer 7

the triple integral for V is
\[ V= \int_0^1 \int_0^{2\pi} \int_0^{\sqrt{1-z}} r\ dr\ dA \ dz \]
here dA is the angle (easier to type than \( d \theta\) )
the inner integral gives
\[ \int_0^{\sqrt{1-z}} r\ dr = \frac{r^2}{2} \bigg|_0^{\sqrt{1-z}}= \frac{1-z}{2}\]
the middle integral gives
\[ \int_0^{2\pi} \frac{1-z}{2}\ dA =\frac{1-z}{2} A \bigg|_0^{2\pi}= (1-z)\pi \]
the outer integral gives
\[ \int_0^1 (1-z)\pi \ dz = \pi\left(z - \frac{z^2}{2}\right)\bigg|_0^1= \frac{\pi}{2}\]
that means
\[ V= \frac{\pi}{2} \]

now evaluate
\[ \frac{1}{V}\int_0^1 \int_0^{2\pi} \int_0^{\sqrt{1-z}} r^2\ z\ r\ dr\ dA \ dz \\
= \frac{1}{V}\int_0^1 \int_0^{2\pi} \int_0^{\sqrt{1-z}} r^3\ z\ dr\ dA \ dz
\]

Answer 8

\[\int\limits_{0}^{1}\frac{ \pi z(1-z)^4 }{ 2 }dz\]

Answer 9

Is expanding only option

Answer 10

That would be the result of the inner and middle integrals if you change it to (1-z)^2

Answer 11

in other words when you sub in sqr(1-z) in for r^4 you get (1-z)^2

Answer 12

ohh ya it is sqrt my bad

Answer 13

and yes, I would expand the binomial

Answer 14

pi/24

Answer 15

so 1/12

Answer 16

yes, I get 1/12
for the average value of f(x,y,z)

Answer 17

btw, here is the lecture where Prof Auroux talks about cylindrical coords
http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-25-triple-integrals/