Famous integral's of exponiental

[dj023102]

Hi guys, i have attached my question. really am stuck with this one. apparently it is a famous integral from poisson. Any ideas on where to start would be good. Cheers

 

[HallsofIvy]

I, and many other people, will not open "Word" files. They are notorious for having viruses.

 

[tiny-tim]

… famous French fish …

Hi guys, i have attached my question. really am stuck with this one. apparently it is a famous integral from poisson. Any ideas on where to start would be good. Cheers

Hi dj023102!

If you really are stuck, then you have plenty of time to type it out for us …

 

[dj023102]

Gee you guys are difficult :)
it's all good just thought it was quicker to use word to type the query up, let's see how i go using latex
evaluate[Infinity ]\int[/-Infinity] [e]^{-x^2}.
does that make sense?
basically integrate e^(-x squared) with upper boundary of infinity and lower boundary of minus infinity.
The tutor wants to image the graph in 3D, the graph of the function
z = e^(-x^2) on the xz-plane. Then rotate this graph around the z-axis to get a bell shaped infinite surface hovering above the xy-plane. The solid chunk is in a space between the xy-plane and the shape looks like a bell.

When you intersect the bell with the infinite hollow cylinder of radius r and the z-axis as its central axis you get a finite hollow cylinder. How do you find the area of this finite hollow cylinder (this cylinder is lacking the caps on both ends, so the area of these caps don’t occur towards the area of the cylinder)

I have no idea where to start, I always thought a cylinder only has volume. Anyone have any ideas where to begin with this one?

 

[Dick]

You want the integral of e^(-x^2)*dx over all x, call it I. That's the same as the integral e^(-y^2)*dy. If you multiply them together you get I^2=integral e^(-(x^2+y^2))dxdy in the x-y plane. You can do this integral by changing to polar coordinates. I think that's what you want to do from your description.

 

[dj023102]

The question ask what is the area of a finite cylinder of radius r when it intersect the bell when the function z=e^(-x^2) is rotated around the z axis (to make the bell).
The area is A = pi times (f(x))^2
Is that what the question is asking i think, but is f(x) = e^(-x^2) or because it is rotated around the z axis i need to make x the subject so f(z) = square root of -In(z)
Does it really matter?

 

[Dick]

The question ask what is the area of a finite cylinder of radius r when it intersect the bell when the function z=e^(-x^2) is rotated around the z axis (to make the bell).
The area is A = pi times (f(x))^2
Is that what the question is asking i think, but is f(x) = e^(-x^2) or because it is rotated around the z axis i need to make x the subject so f(z) = square root of -In(z)
Does it really matter?

You find the area of that cylinder just like you find the area of any other cylinder. It's the height times the circumference.

 

[dj023102]

Yeah but which area should i use?

f(x) = e^(-x^2) or because it is rotated around the z axis i need to make x the subject so f(z) = square root of -In(z)

cos after that i need to calculate the volume of the cylinder by integrating the area with lower bound 0 and upper bound infinity. How would i go about using that? but firsty i need to know which function to use, any help there?

 

[Dick]

Once you rotate it x^2 becomes r^2=x^2+y^2. So the height of the cylinder is exp(-r^2). What's the radius? You are overthinking this.

 

[dj023102]

Ok the height of the exp(-r^2) is, but what about the area? that's is what i having the problem with. i am assuming that i should take the area of the circular disk and times it by the height (exp(-r^2)) and that will give me the volume.

But what is the area i should be multiplying by?

 

[Dick]

You want the AREA of the CYLINDER. You will later integrate that to get the volume of the solid. You don't want the volume of the cylinder. Again, area=height times circumference.

 

[dj023102]

ah ok! i think i get it. Sorry been looking at the wrong part of the text.
So area of cylinder = (exp(-r^2))*(2*r*pi).
Is that right?

 

[splatter]

hey I've been watching this thread for a while, and I am pretty sure i have exactly the same assignment.. anyway the one i have is broken up into 6 sections, the 4th of which is really bugging me. It says "express the area of intersection of the bell (the rotated integral) with the plane parallel to, and a distance y, from the xz-plane as a function S(y).
just stuck on how you would put this intersection into writing, or where to start from. any ideas?

 

[dj023102]

hang on there i need more help, stuck on the first part of the question. but i am heading in the right direction now am i?

 

[splatter]

yeah I am pretty sure, that's how i did that part anyway. if you've found C(r) it should be fairly easy to integrate.

 

[splatter]

hey I've been watching this thread for a while, and I am pretty sure i have exactly the same assignment.. anyway the one i have is broken up into 6 sections, the 4th of which is really bugging me. It says "express the area of intersection of the bell (the rotated integral) with the plane parallel to, and a distance y, from the xz-plane as a function S(y).
just stuck on how you would put this intersection into writing, or where to start from. any ideas?

i think i have that part now, but there's some other stuff that follows on from that which is really confusing me, even though it should be fairly straight forward. i THINK i have S(y), and it then says that the integral S(y) over all y is equal to the volume of the bell, given by the integral of the cylinder mentioned earlier in thread. S(y) ends up as "integral e^(-(x^2+y^2))dx over all x" i think, but it then says to find the integral of S(y) over all y, which will contain two integral signs, and separate integral S(y) into the product of two simple integrals, one containing only variable x and the other containing only y. the reason I am confused is that i end up with [integral e^(-x^2)dx over all x]*[integral e^(-y^2)dy over all y], which presents the same problem as the initial integral of I=int e^(-x^2) dx over all x...

 

[tiny-tim]

Hi dj023102!

(have an integral: ∫ and a squared: ² and a theta: θ )

z = e^(-x^2) on the xz-plane. Then rotate this graph around the z-axis to get a bell shaped infinite surface hovering above the xy-plane. The solid chunk is in a space between the xy-plane and the shape looks like a bell.

When you intersect the bell with the infinite hollow cylinder of radius r and the z-axis as its central axis you get a finite hollow cylinder. How do you find the area of this finite hollow cylinder (this cylinder is lacking the caps on both ends, so the area of these caps don’t occur towards the area of the cylinder)

I have no idea where to start, I always thought a cylinder only has volume. …

cos after that i need to calculate the volume of the cylinder by integrating the area with lower bound 0 and upper bound infinity.

hmm … you might have said that at the beginning, so that we knew what the context was.

ok … the aim is to find the volume under the bell.

To find a volume, you're probably used to dividing the volume into horizontal slices of thickness dz, and then integrating over z.

But in cases of rotational symmetry (like this), it's easier to use cylindrical slices (or "shells") of thickness dr, and then integrate over r.

The volume of a cylindrical shell is A dr, where A is the area …
that's the only reason you need to know the area!

… the reason I am confused is that i end up with [integral e^(-x^2)dx over all x]*[integral e^(-y^2)dy over all y], which presents the same problem as the initial integral of I=int e^(-x^2) dx over all x...

Hi splatter!

Hint: ∫∫e^-x² e^-y² dx dy = ∫∫e^-r² dx dy …

now change the variables of integration from x and y to r and θ.

 

[Dick]

ah ok! i think i get it. Sorry been looking at the wrong part of the text.
So area of cylinder = (exp(-r^2))*(2*r*pi).
Is that right?

That's exactly right. If you integrate that r=0 to infinity you get the volume of the region under the bell. And, as splatter is finding, that is I^2, where I=integral exp(-x^2)dx.

 

[splatter]

hmm … you might have said that at the beginning, so that we knew what the context was.

Sorry about that :P i see the stupidity now.. hehe
thanks heaps for that though - big help! :) i hope i can do it properly, its been making me crazy!

Thanks again

 

[splatter]

Hint: ∫∫e^-x² e^-y² dx dy = ∫∫e^-r² dx dy …

now change the variables of integration from x and y to r and θ.

sorry i just realized i don't understand what you meant by changing the variable y to θ - i don't have any variables except x, y, r, and possibly z.. sorry to be so annoying =/

 

[tiny-tim]

r and θ

sorry i just realized i don't understand what you meant by changing the variable y to θ - i don't have any variables except x, y, r, and possibly z.. sorry to be so annoying =/

Hi splatter!

I meant the usual polar (or cylindrical) coordinates r and θ:

x = rcosθ, y = rsinθ, ∫∫ dx dy = ∫∫ r dr dθ.

 

[splatter]

ahh yep yep, i was heading in that direction but wasn't too sure :P

thanks again tiny-tim!

 

[His_Dudeness3]

Okay, I was handed an assignment like this on Friday and I also have no clue.

For the question:

Asking for the area of 'this finite cylinder', C(r), I got: C(r)= [exp]^(-r^2)* 2*pi*r

Asking for the volume of the bell, i.e. the integral of C(r)dr with the upper limit being positive infinity and the lower limit being zero, I got: V= -pi*([exp]^(+infinity)-1)

Now I have gone over all my lecture notes, and I seem to have no clue how to do the rest of the questions. Can someone please help me?

(1) Show that the top of the bell is the graph of the function:
z=[exp]^(-(x^2+y^2))

(2) Express the area of the intersection of the bell with the plane parallel to, and a distance y from the xz-plane as a function S(y). Note that for y=0 we find that S(y) is equal to our mystery integral, I. In general, S(y) is just like the special case S(0), a definite integral. Now it turns out that the volume of the bell is also equal to

[Integral](-infinity,+infinity) S(y)dy

(3) This second expression for the volume contains two integral signs. Manipulate it until it looks as follows.
[Integral](-infinity,+infinity)f(x)*([integral](-infinity,+infinity)g(y)dy) dx, with suitable functions f(x) and g(y).

This 'double' integral is equal to:
[Integral](-infinity,+infinity)f(x)dx*[Integral](-infinity,+infinity)g(y)dy, the product of two simple integrals.

(4) Equate the two expressions for the volume of the bell to find our mystery integral I.

Any help is greatly appreciated.

 

[tiny-tim]

Welcome to PF!

Hi His_Dudeness3! Welcome to PF!

(have a pi: π and an infinity: ∞ and an integral: ∫ )

erm … let's just start with this:

Asking for the volume of the bell, i.e. the integral of C(r)dr with the upper limit being positive infinity and the lower limit being zero, I got: V= -pi*([exp]^(+infinity)-1)

Do you mean V = -π(e^+∞ - 1)?

that's ∞ (which is wrong).

Look at it again … I think you'll find you got a minus in the wrong place.

 

[His_Dudeness3]

Hey Tiny-tim, thanks for the welcome. I look forward to begging you, and other posters for help :). Anyway, yeah I got the answer for the volume as V = -pi(e+∞ - 1) .
I'm looking very hard and I don't know what mistake I've done. Is it supposed to be the other way around?

 

[tiny-tim]

Hi His_Dudeness3!

(what happened to that π and ∞ I gave you?)

I'm looking very hard and I don't know what mistake I've done.

erm … nor do I, until you show us what you did!

That's the rule on this forum … show us your full calculation, then we can see what the problem is.

_{(my guess would be you integrated e}r² _{instead of e}-r²₎

 

[His_Dudeness3]

Hi His_Dudeness3! Welcome to PF!

(have a pi: π and an infinity: ∞ and an integral: ∫ )

erm … let's just start with this:


Do you mean V = -π(e^+∞ - 1)?

that's ∞ (which is wrong).

Look at it again … I think you'll find you got a minus in the wrong place.

ohh sorry, lol do you mean its supposed to read as follows:

V= -pi*( [exp]^(-∞) -1)

 

[His_Dudeness3]

Okay, I tried that 'find the volume of the cylinder question again:

1. First I Integrated C(r), the Area of the bell over the xz plane, to get:

V= -pi [ [exp]^(-r^(2)) - 1 ],upper boundary=∞, lower boundary=0

2. The next question asks, show that the top of the bell is the graph of the function:
z=[exp]^(-(x^2 + y^2))

I thought I could get this by getting the derivative of V, and finding the top of the bell (i.e. the maximum) by making dV/dr = 0 but I ended up getting -r^(2) = log(o)/log([exp])

Anyone got any thoughts? Am I on the right track?

 

[tiny-tim]

ohh sorry, lol do you mean its supposed to read as follows:

V= -pi*( [exp]^(-∞) -1)

Hi His_Dudeness3!

Why not go on to say [exp]^(-∞) = 0 (because it's 1/[exp]^(+∞), = 1/∞, = 0 ), so V = π?

2. The next question asks, show that the top of the bell is the graph of the function:
z=[exp]^(-(x^2 + y^2))

I thought I could get this by getting the derivative of V, and finding the top of the bell (i.e. the maximum) by making dV/dr = 0 but I ended up getting -r^(2) = log(o)/log([exp])

Anyone got any thoughts? Am I on the right track?

Nooo … the question isn't clear, and you've misunderstood it.

The bell is the volume, and the "top" of the bell is its top surface …

which is z=[exp]^(-(x^2 + y^2)), isn't it?