Volume of Polar Solids with Cross Sections

[smittytibbs]

Homework Statement

I'm trying to find the volume of a Polar shape with semi-circular cross sections. Since it is a polar graph, does that mean the cross sections are to be swept across the graph from 0 to 2π in triangular sections?
I'm aiming to create one side of a three dimensional nautilus shell by superimposing these two polar graphs: r(t) = 3 + cos (t/2) and r(t) = 1 + cos (t/2) from [0, 2π].

Could someone help me understand and explain the process involved in finding the volume please?

Homework Equations

The Attempt at a Solution

I'm pretty sure it won't work if I just integrate regularly (by integrating the area of the cross section with the radius equaling the outer polar equation minus the inner polar equation). I'm only a junior in calc BC, and i think the concept/math involved is beyond that. I'm making a model, in which the actual volume can be calculated to scale, but i just don't know about the math part.

 

[mighty2000]

Hello there,

I understand your confusion in finding the volume of a polar shape with semi-circular cross sections. Let me explain the process involved in finding the volume.

Firstly, you are correct in assuming that the cross sections will be swept across the graph from 0 to 2π in triangular sections. This means that the cross sections will be in the shape of a triangle, with two sides being the semi-circular arcs and the third side being the radius of the polar graph.

To find the volume, we need to use the formula for the volume of a solid of revolution, which is given by:

V = ∫πr^2dx

In this case, r represents the radius of the polar graph at a particular angle, and dx represents the infinitesimal angle from 0 to 2π.

To apply this formula to your specific problem, you will need to integrate the difference between the outer polar equation (3 + cos (t/2)) and the inner polar equation (1 + cos (t/2)) squared, from 0 to 2π.

So the final integral will be:

V = ∫0^2π(3 + cos (t/2))^2 - (1 + cos (t/2))^2 dt

After integrating and simplifying, you will get the volume of the nautilus shell.

I hope this helps you understand the process involved in finding the volume of a polar shape with semi-circular cross sections. Let me know if you have any further questions.