Volume of complex shapes

[MTP27]

I have a shape that I made as a final project in Calculus. We were doing known cross sections and had to do a fairly complex base with a simple cross section. I didn’t do that.

What would I do for a cross section that changes area depending on the length of he base.

My shape looks something like this

Or if I found a way to calculate the surface area I would be able to double integration.

This project is almost a year old so any help would be great.

 

[Dr.Peterson]

Just integrate the cross-sectional area. It doesn't have to be a simple shape to do that (though it does have to be defined clearly) -- most volumes don't have constant cross-sectional area. Finding the areas in this case will presumably require integration; so this really amounts to double integration.

Please show what you've tried. At least define the cross-sections, and show an attempt at an integral.

Of course, not all integrals can be carried out exactly; that will be a separate issue.

 

[MTP27]

This picture is the bounds of my cross section. I can also put a link to my Desmos graph of the section at the bottom

Because there are three different cases with the upper function I had to define where it switched from one to another

The bottom function is what I assume to be the equation for the volume with indexing vectors for the area for each of the cases


This is my shape on the bottom where the Base of the section is top to bottom and I’ve declared it as ‘s’. S goes from 0 to 12


My cross section areas is defined by the red and black overlap and as the slider ‘s’ changes so does the cross section. On the first picture, I’ve declared the two vectors A and B where all of the numbers are where the case switched from one type to another. I’ve extended out the decimals so that my answer Is correct our to the 5th decimal

https://www.desmos.com/calculator/sshul6botm

Thanks again for the help

 

[Dr.Peterson]

I have to admit I have no interest in pursuing this at all. It seems like a huge waste of time. But somebody else may want to help.

 

[MTP27]

No that’s totally fine. I understand. My interest has been renewed recently so I’ve decided to try looking again.

Thanks for looking either way