Volume of intersection of two perpendicular solid cylinders

Calculus Level 4

Two solid circular cylinders of equal radius 3 cm 3\text{ cm} are intersecting each other perpendicularly (As shown in the figure below). Find out the volume of solid of intersection in cm 3 \text{cm}^3 .


The answer is 144.

Harish Chandra Rajpoot by Harish Chandra Rajpoot , 30, India

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1 solution

In general, the equations of intersecting cylinders of radius R R can be taken as x 2 + y 2 = R 2 x^2+y^2=R^2 & x 2 + z 2 = R 2 x^2+z^2=R^2 . Now, using double or triple integration one can easily derive the formula of volume of intersection of solid cylinders as follows

V = 16 3 R 3 V=\frac{16}{3}R^3

setting the value of radius R = 3 c m R=3\ cm one can find the volume of intersection

V = 16 3 ( 3 ) 3 = 144 c m 3 V=\frac{16}{3}(3)^3=144\ cm^3

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