Volume Formula for Ellipsoid (a,b,c > 0) using Calculus: Slicing and Rotation

[vip89]

Can you help me , please??

Can you solve this problem by using Calculus l "" Volumes by Slicing and Rotation About an Axis"" ??
Develop a formula for the volume of an ellipsoid of the form

( x^2/a^2)+(y^2/b^2)+(z^2/c^2)=1, a, b, c > 0.

I want the steps , because I know the final answer .

 

[tiny-tim]

Welcome to PF!

Hi vip89! Welcome to PF!

Show us what you've tried, and where you're stuck, and then we'll know how to help you!

 

[vip89]

I know that I must sketch,find the equation of the cross section , find the area from A=piab and x^2/a^2+y^2/b^2=1
Finally,I must find the definit integral of the area
the final answer will be 4pi/3 abc

 

[vip89]

I don't know how to do this , and how to apply my steps?!

 

[tiny-tim]

Hi vip89!

I know that I must sketch,find the equation of the cross section …

ok … we usually take horizontal cross-sections (don't have to) …

in other words, the intersection with a plane z = constant.

So what is the equation (in x and y) if you put z = w (a constant) … ?

… and then what is the area?

 

[vip89]

thnkx very much

 

[vip89]

I trid to solve it by this way,but strang things will appear,the equation be more complicated!

 

[Tedjn]

Show us your work. You can either take a picture, scan it, and upload it, or you can try using LaTeX in between [noparse]and[/noparse] tags.

 

[tiny-tim]

I trid to solve it by this way,but strang things will appear,the equation be more complicated!

Hi vip89!

Show us how far you got …

What equation (in x and y) did you get for the horizontal cross-sections when you put z = w (a constant)?

 

[rocomath]

I want the steps , because I know the final answer .

AND WE WANT YOUR WORK!

lol ... funny kid we're not here to do your hw

 

[tiny-tim]

Welcome to this thread, rocomath!

oi … rocomath! … that was vip89's very first post (seven days ago)!

… but he's getting the hang of it now!

If you can't help, just say "welcome!" ​

 

[rocomath]

oi … rocomath! … that was vip89's very first post (seven days ago)!

… but he's getting the hang of it now!

If you can't help, just say "welcome!" ​

sorry :( I've grown less tolerant towards ppl just wanting us to do their hw.

 

[vip89]

These are my trials
I have the idea but can't get the right answer
pls reply
THNKX

 

[vip89]

Continue:

 

[HallsofIvy]

In your first sheet, you start with \int A(x)dx and then immediately switch to \int x y[/itex] without a &quot;dx&quot; at all. How did that happen? I also cannot see any good reason for writing x as a function of y and z <b>and</b> writing y as a function of x and z.<br /> You are correct that y= b\sqrt{1- x^/a^2- z^2/c^2} and that z= c\sqrt{1- x^2/a^2- y^2/b^2. The first tells you that when z= 0, y= b\sqrt{1- x^2/a^2} and the second tells you that when y= 0, z= c\sqrt{1- x^2/a^2}. In other words, at each x, the cross section is an ellipse with semi-axes b\sqrt{1-x^2/a^2} and c\sqrt{1- x^2/a^2}. Do you know that the area of an ellipse with semi-axes a and b is \pi ab?

 

[tiny-tim]

hmm … let's start at the top of your page 3 …

w² = (1 - x²/a² - y²/b²)c²

which I take it is following my suggestion …

So what is the equation (in x and y) if you put z = w (a constant) … ?

… and then what is the area?

And then you try to use the formula A = πab for the area of an ellipse.

BUT … that formula only applies if the equation for the ellipse is in the standard form, with nothing but x² and y² on the left and "= 1" on the right.

You must put w² = (1 - x²/a² - y²/b²)c² into that form first.

Then you will get a "new a and b" that are not the same as the original a and b.

ok … rearrange w² = (1 - x²/a² - y²/b²)c² into the standard form … and then find the area.