Volume By Cross Section

[sisxixon]

Here's the problem:
The axes of two right circular cylinders of radius a intersect at right angles. Find the volumem of the solid bounded by the cylinders.

The problem is, i'm not sure what this solid looks like to even set up the integral.

What i think the solid looks like is one cylinder lying along its height on the x axis, with one base on the y axis and the other base at x=h. the other cylinder would be vertical, with its base on the x axis and its height up the y axis, from y=0 to y=h

After a bit of messing around, what i did was this:

square cross sections

s=2a

A=4a^2

V=integral(from 0 to 2a) 4a^2dx=8a^3

could someone please check over my work? I'm not even sure if i've got the solid right, so ifyou could kindly point me in the right direction, it would be much appreciated.

 

[galactus]

Cool problem.

If we use an octant, find its volume and multiply by 8, that may be a good

way to go about it.


Each cross-section perpendicular to the y-axis is a square. See?.

So, we have \(\displaystyle A(y)=x^{2}=a^{2}-y^{2}\)

Therefore, \(\displaystyle V=8\int_{0}^{a}(a^{2}-y^{2})dy\)

Integrate to find volume of region bounded by their intersection.

 

[sisxixon]

oh, i got it! many thanks

 

[Snowdog2112]

I have the same basic problem, but with one change, "Find the volume of the solid that is bounded by two right circular cylinders of radius r, if their axis meet at angle θ.". How are you supposed to do this one?.

 

[stapel]

I have the same basic problem, but with one change, "Find the volume of the solid that is bounded by two right circular cylinders of radius r, if their axis meet at angle θ.". How are you supposed to do this one?

Please post new questions as new threads, not as "hijacks" of other students' threads.

Thank you.

Eliz.