Integrating a specific integral based on the volume of the intersection of two cylinders. V=int_-r^r sqrt{r^2-y^2} sqrt{R^2-y^2}dy. Where r leq R. If b=R/r, how can I show V=r^3F(b).

chemicars8

chemicars8

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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

Answer & Explanation

gypePlealeLertv

gypePlealeLertv

Beginner2022-08-26Added 9 answers

Step 1
If you let u = y / r, you get the integral
V = 1 1 r 2 r 2 u 2 R 2 r 2 y 2 r d u = r 3 1 1 1 u 2 R 2 / r 2 u 2 d u .
Step 2
Since b = R / r, we find that V = r 3 F ( b ), with
F ( b ) = 1 1 1 u 2 b 2 u 2 d u .
Note that we do not need to calculate F explicitly.

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