Are you interesting about average value if so trying here.

[memomath]

Hello

here you have the same question in the enclosed

What is the average value?

average value for (exp $$\alpha$$Z ) = $$\int$$$$\int$$ exp$$\alpha$$Z ds / $$\int$$$$\int$$ ds



Over the sphere S: X^2+Y^2+Z^2=a^2

Also by use the parameterization

X= a sinϴ cos ɸ
Y= a sinϴ sin ɸ
Z= a cos ϴ

And the usual substitution t = cos ϴ in the integral:

$$0\int$$$$\Pi$$ f (cos ϴ) sinϴ dϴ =$$-1\int$$1 f(t) dt

Thanks

 

[memomath]

No comments or solution until now!

hello

please open the enclosed to find a nice problem and to solve this interesting question

thanks

 

[HallsofIvy]

Again, Word files are notorious for harboring viruses. I will not open one from someone I do not know.

 

[memomath]

Hello here the same question in the enclosed

What is the average value?

average value for (exp $$\alpha$$Z ) = $$\int$$$$\int$$ exp$$\alpha$$Z ds / $$\int$$$$\int$$ ds



Over the sphere S: X^2+Y^2+Z^2=a^2

Also by use the parameterization

X= a sinϴ cos ɸ
Y= a sinϴ sin ɸ
Z= a cos ϴ

And the usual substitution t = cos ϴ in the integral:

$$0\int$$$$\Pi$$ f (cos ϴ) sinϴ dϴ =$$-1\int$$1 f(t) dt

Thanks

 

[0xDEADBEEF]

I can kind of guess what you mean, but your notation is pretty bad. I do not know what exp$$\:^\alpha Z$$ is supposed to be. Your substitution seems wrong, your notation is unorthodox and you write factors like -1 instead of just a - . This is probably the reason why people don't answer.

You might have better chances if you wrote something like:

"I am trying to get the average $$\left< f(z) \right>_{S_a}$$ of a function $$f(z)= \alpha ^ z$$ over a sphere $$S_a$$ of radius $$r=\sqrt{a}$$. I swear this is not for homework.

What I have so far is this:

$$\left< f(z) \right>_{S_a} = \frac{\int_{S_a} \alpha^z\,\mathrm{d}\Omega}{\int_{S_a} \,\mathrm{d}\Omega}$$

I tried to express the integral in polar coordinates:
X= a sinϴ cos ɸ
Y= a sinϴ sin ɸ
Z= a cos ϴ
$$\theta \in \left[0,\pi \right]$$
$$\phi \in \left[0,2\pi \right]$$

and found:
$$\int_{S_a} f(z) \,\mathrm{d}\Omega = \int_0^{\pi} f(\cos \theta) \sin \theta \, \mathrm{d}\theta$$

Is this correct? How do I proceed?"

Then people would have told you what you did wrong.

 

[memomath]

Thanks for reply but it is not correct hint and I think you don’t proceed with the right way for the real problem you probably fix some thing else many things missing from the real posted thank you

I can kind of guess what you mean, but your notation is pretty bad. I do not know what exp$$\:^\alpha Z$$ is supposed to be. Your substitution seems wrong, your notation is unorthodox and you write factors like -1 instead of just a - . This is probably the reason why people don't answer.

You might have better chances if you wrote something like:

"I am trying to get the average $$\left< f(z) \right>_{S_a}$$ of a function $$f(z)= \alpha ^ z$$ over a sphere $$S_a$$ of radius $$r=\sqrt{a}$$. I swear this is not for homework.

What I have so far is this:

$$\left< f(z) \right>_{S_a} = \frac{\int_{S_a} \alpha^z\,\mathrm{d}\Omega}{\int_{S_a} \,\mathrm{d}\Omega}$$

I tried to express the integral in polar coordinates:
X= a sinϴ cos ɸ
Y= a sinϴ sin ɸ
Z= a cos ϴ
$$\theta \in \left[0,\pi \right]$$
$$\phi \in \left[0,2\pi \right]$$

and found:
$$\int_{S_a} f(z) \,\mathrm{d}\Omega = \int_0^{\pi} f(\cos \theta) \sin \theta \, \mathrm{d}\theta$$

Is this correct? How do I proceed?"

Then people would have told you what you did wrong.

 

[girlmatrix]

YES memomath I believe there are some mistakes and the good hint to the correct answer for this problem would be in ENGINEERING MATHEMATICS JOHN BIRD

Thanks for reply but it is not correct hint and I think you don’t proceed with the right way for the real problem you probably fix some thing else many things missing from the real posted thank you