Finding the volume of an object with 3 parameters. given an equation of ellipsoid, it has 3 parameters: x^2+4y^2+4z^2 leq 4

Alexus Deleon 2022-09-15 Answered
Finding the volume of an object with 3 parameters
I know how to find the volume of a sphere/ball around x-axis using:
V = π a b f 2 ( x ) d x
Lets say if:
x 2 + y 2 = r 2
We do: y = r 2 x 2
So now: V = π r r ( r 2 x 2 ) 2 d x
But the problem starts here:
Im given an equation of ellipsoid, it has 3 parameters:
x 2 + 4 y 2 + 4 z 2 4
What do i do with the z parameter? How do i build y now? how does it fit to the equation by integrals of V?
I would like an explanation and not just a solution - because its homework.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

ticotaku86
Answered 2022-09-16 Author has 7 answers
Step 1
In finding the volume of a ball x 2 + y 2 + z 2 r 2 , you revolve the curve y = r 2 x 2 about the y-axis. Notice that y = r 2 x 2 is the equation you get by setting z = 0 in the equation for the sphere.
You can do the same thing with the ellipsoid: set z = 0 to get the equation for the boundary of the rugby ball in the (x,y)-plane, and solve for y.
x 2 + 4 y 2 = 4 y = 4 x 2 2
Step 2
Revolving this curve about the y-axis gives the volume,
π 2 2 ( 4 x 2 2 ) 2 d x = π 4 2 2 4 x 2 d x = 8 π 3

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-08-31
Finding the volume of a cube using spherical coordinates
Calculate the volume of a cube having edge length a by integrating in spherical coordinates. Suppose that the cube have all the edges on the positive semi-axis. Let us divide it by the plane passing through the points (0;0;0),(0;0;a),(a;a;0).
We get two equivalent prism; now we divide the section again by the plane passing through the points (a;0;a),(0;0;0),(a;a;a); So one may write:
1 2 V = ( 0 π / 4 d ϕ 0 π / 4 d θ 0 a cos ϕ r 2 sin ϕ d r + π / 4 π / 2 d ϕ 0 π / 4 d θ 0 a sin ϕ cos θ r 2 sin ϕ d r )
So one should expect V = a 3 but the latter expression gives a different result. What is the correct way to calculate V and why doesn't my reasoning work?
asked 2022-09-19
Finding volume of a sphere using integration
I have searched and found 2 methods of finding volume using integration:
- considering a small cylindrical element and integrating that over the radius
- considering a small circle element and using the relation x 2 + y 2 = r 2 and integrating it over the z-axis.
I was trying to find the integration by considering a small circle element (with radius r) and using the relation r = R cos θ where R is the radius of the sphere / hemisphere.
So I was thinking of calculating the volume of the hemisphere by integrating the π R 2 cos 2 θ d θ from θ to π / 2. Is this method right? And how will the integration be like?
asked 2022-08-16
Multivariable calculus double integration volume question
Use a double integral to find the volume of the solid bounded by graphs of the equations given by:
z = x y 2 ,  where:  z > 0 x > 0 5 x < y < 2
My problem is finding the limits of integration. I know my f ( x , y )... my guess is my y integration is going from 5x going to 2. But how do I find the x limitations?
asked 2022-07-22
Finding the volume bounded by surface in spherical coordinates
R = 4 1 cos ( ϕ )
asked 2022-07-26
a cylinder whose height is 8 inches has a volume of 128 pi cm3. if the radius is doubled and its height is cut in half, the volume of the resulting cylinder is...
please show how to do it i do not need the answer i know the answer is 256pi cm^3
asked 2022-08-25
Integrating a specific integral based on the volume of the intersection of two cylinders
I have been able to obtain the formula to calculate the volume like the solution to the link above.
My formula looked like this:
V = r r r 2 y 2 R 2 y 2 d y
Where r R
if b = R r , how can I show V = r 3 F ( b )
I am having trouble finding an expression where volume is equal to r 3 times some expression that is only dependant on b
asked 2022-08-16
Finding volume of pyramid in R 3 .
Assume we have a pyramid (triangular base) sitting in R 3 , i.e. we have the coordinates for the vertices A,B,C and D. How can we find the volume of the pyramid without using either its height or its base area and no advanced tools like integration?

New questions