Volume Integration Using Cylindrical Polar Coordinates

[DanAbnormal]

Homework Statement

Hi guys. Just before I ask I want to tell you my situation.
Im doing a course in Astronomy in London, and as much as i'd say I was pretty good at physics and am highly interested unfortunatley maths is a turbulant subject for me. Unsuprisingly I didnt do too well in my first year finals with respect to maths, and failed my two maths modules marginly.
Because of this they sent me the papers to do at home, to be follwed up by an interview to see if I am capable of moving to second year.
Only thing is I have to complete every question, whereas on the exam day there was an option to choose 5 out of 7 blah blah.
So now I am just having quarms with stuff I left out that I literally could not understand. I just want people to know I might have quite a few questions on this forum, and id like to apologise in advance if it just looks like I am taking advantage, but I've done most of these papers now, all I have left is some things I don't get, which I have tried to find a solution myself, but to no avail.

So here's my first question, couldn't do it in the exam.

A giant parabolic space dome has a roof described by the equation

z = 1 - x^2 - y^2

where z is the height above the ground and x and y are horizontal coordinates all measured in km.

Calculate the volume of the dome, using cylindrical polar coordinates.

There's more of the question, but i'll see if I can do it with a solution to this part.

Homework Equations

Well I am quite aware that

x = rcos0
y = rsin0
z = z

So the volume element is dv = rdrd0dz

The Attempt at a Solution

But in all honesty I do not know where to start, we never did any work on dome structures during term time. But I do understand the process of integration by polar coordinates so I won't need an explanation for that.

Thank you.

 

[Defennder]

I assume the volume of the region is bounded below by the x-y plane. Think in terms of a double integral over a region dA. How would you find the differential volume there? Setup the integral in rectangular coordinates first, then convert to polar and solve it.

 

[DanAbnormal]

I don't understand...

The equation z = 1 - x^2 - y^2 describes the roof of the equation, so do I integrate that equation?
If that is the case then I am unsure what limits to use.

If I look at it I think the limits for the dz element are 1 - x^2 - y^2 and zero. But not sure how that works if that's the equation I am integrating...

 

[Dick]

I don't understand...

The equation z = 1 - x^2 - y^2 describes the roof of the equation, so do I integrate that equation?
If that is the case then I am unsure what limits to use.

If I look at it I think the limits for the dz element are 1 - x^2 - y^2 and zero. But not sure how that works if that's the equation I am integrating...

Yes, you are integrating dV over the volume, and yes, the dz limits are 0 -> 1-x^2-y^2. So this gives you a factor of (1-x^2-y^2). Now you just want to integrate that r*dr*dtheta, right?

 

[merryjman]

I remember a technique from intro calculus, where (for example) one would calculate the volume of a frustum of a cone by first creating a dV volume element for a thin disk, then adding up all these disks. The only tricky part is that the width of the disks changed for each dV, so that's why integration was needed.

Here, you've got a similar situation. As someone else mentioned, your building is bounded on the bottom by the xy plane, and on the top by the equation you wrote. Change your variables around and set up the correct integral. Most calculus texts have examples of this kind of thing.