Using Symmetry for finding volume I have a confusion regarding the symmetry of the volume in the following question. Find the volume common to the sphere x 2 + y 2 + z 2 = 16 and cylinder x 2 + y 2 = 4 y . The author used polar coordinates x = r c o s θ and y = r s i n θ and does something like this: Required volume V = 4 ∫ 0 π / 2 ∫ 0 4 s i n θ ( 16 − r 2 ) 1 / 2 r d r d θ . The reason for multiplying by 4 is the symmetry of the solid w.r.t. xy-plane. My point of confusion is that this solid cannot be cut into 4 identical parts, so how it can be multiplied by 4?

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Using Symmetry for finding volume    I have a confusion regarding the symmetry of the volume in the following question.    Find the volume common to the sphere           x      2        +          y      2        +          z      2        =    16   and cylinder           x      2        +          y      2        =    4    y  .    The author used polar coordinates     x    =    r    c    o    s    θ   and     y    =    r    s    i    n    θ   and does something like this:    Required volume     V    =    4          ∫      0                        π                          /                2                    ∫      0              4        s        i        n        θ              (    16    −          r      2              )              1                  /                2              r    d    r    d    θ  . The reason for multiplying by 4 is the symmetry of the solid w.r.t. xy-plane.    My point of confusion is that this solid cannot be cut into 4 identical parts, so how it can be multiplied by 4?

Calvin Maddox · Accepted Answer

Step 1My point of confusion is that this solid cannot be cut into 4 identical partsSure it can.The sphere is centered at the origin, and the axis of the cylinder (which has radius 2) intersects (0,2) in the   x  −  y plane.Step 2Slice it in half with the   z  =  0 plane. Then slice it again with the   x  =  0 plane.These are reflected in the limits of integration.

Jadon Johnson · Accepted Answer

Explanation:The volume can be cut into 4 identical parts. See that both volume are simetric for each quadrant. That&#039;s why the angle variation is from 0 to   π      /    2.

Find the volume common to the sphere x^{2}+y^{2}+z^{2}=16 and cylinder x^{2}+y^{2}=4y.

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