Integrating a specific integral based on the volume of the intersection of two cylindersI 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 yWhere r ≤ Rif 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

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Integrating a specific integral based on the volume of the intersection of two cylindersI 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  yWhere   r  ≤  Rif   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

gypePlealeLertv · Accepted Answer

Step 1If 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 2Since   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.

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).

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