Intuitive error in finding Volume of sphere using single integration.To calculate the volume of a sphere of radius R, I considered a thin disc of radius r = R s i n θ, where θ is the angle radius vector on the circumference makes with the axis of the disc. However I considered the volume d V = π r 2 R d θ, R d θ being the infinitesimal thickness of the disc, which is erroneous. The volume of the sphere comes out to be 3 π 2 R 3 2 . The correct thickness is R s i n θ d θ which is found usually in other derivations gives the correct volume. However I do not understand what exactly is the issue in my approach and why the component of the curve length is considered.

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Intuitive error in finding Volume of sphere using single integration.To calculate the volume of a sphere of radius R, I considered a thin disc of radius   r  =      R    s  i  n  θ, where   θ is the angle radius vector on the circumference makes with the axis of the disc. However I considered the volume   d  V  =   π      r    2        R    d  θ,       R    d  θ being the infinitesimal thickness of the disc, which is erroneous. The volume of the sphere comes out to be             3              π        2                              R                3              2  . The correct thickness is       R    s  i  n  θ  d  θ which is found usually in other derivations gives the correct volume. However I do not understand what exactly is the issue in my approach and why the component of the curve length is considered.

Riya Cline · Accepted Answer

Step 1Let the x-axis of                     R              3   be the axis of the disc. The disc is then bounded by the sphere of radius R and two planes   x  =  a and   x  =  b. We now have to express a and b in terms of   θ and   d  θ. When   b  =  R  cos  ⁡  θ then   a  =  R  cos  ⁡  (  θ  +  d  θ  )  &amp;lt;  b.Step 2The thickness of the disc is then given by   b  −  a  =  R            (        cos  ⁡  θ  −  cos  ⁡  (  θ  +  d  θ  )            )        ≈  R            (        −      cos    ′    ⁡  (  θ  )            )          d  θ  =  R  sin  ⁡  θ    d  θ     ,and has nothing to do with the arc length   R    d  θ along the meridian of B. We obtain       v    o    l    (  B  )  =      ∫    0    π    π      r    2    (  θ  )    R  sin  ⁡  θ    d  θ  =                    4        π            3            R    3       .

To calculate the volume of a sphere of radius R, I considered a thin disc of radius r-R sin theta, where theta is the angle radius vector on the circumference makes with the axis of the disc.

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