Finding volume of a sphere using integrationI have searched and found 2 methods of finding volume using integration:- considering a small cylindrical element and integrating that over the radius- considering a small circle element and using the relation x 2 + y 2 = r 2 and integrating it over the z-axis.I was trying to find the integration by considering a small circle element (with radius r) and using the relation r = R cos ⁡ θ where R is the radius of the sphere / hemisphere.So I was thinking of calculating the volume of the hemisphere by integrating the π R 2 cos 2 ⁡ θ d θ from θ to π / 2. Is this method right? And how will the integration be like?

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Finding volume of a sphere using integrationI have searched and found 2 methods of finding volume using integration:- considering a small cylindrical element and integrating that over the radius- considering a small circle element and using the relation       x    2    +      y    2    =      r    2   and integrating it over the z-axis.I was trying to find the integration by considering a small circle element (with radius r) and using the relation   r  =  R  cos  ⁡  θ where R is the radius of the sphere / hemisphere.So I was thinking of calculating the volume of the hemisphere by integrating the   π      R    2        cos    2    ⁡  θ  d  θ from   θ to   π      /    2. Is this method right? And how will the integration be like?

booyo · Accepted Answer

Step 1The best way of solving this problem it&#039;s using spherical coordinates. Then, you can only calculate the volume of one octant of the space supposing that the sphere is centered on the origin.Step 2So, given a solid sphere with radius R, the volume would be:  V  =  8      ∫          0                      π        2                  d    θ        ∫          0                      π        2                        ∫              0                    R                            r                  2                    sin      ⁡              φ                        d      r        d    φ

Krish Crosby · Accepted Answer

Step 1I do not like the &quot;consider a small element&quot; approach physicists often use, as it is not very intuitive and can easily produce errors. In this case you can just use the usual transformation of integrals:      ∫          Ω        f  (  y  )  d  y  =      ∫                  φ                  −          1                    (      Ω      )        f  (  φ  (  x  )  )      |    det  D  φ  (  x  )      |    d  xStep 2Where   φ  :  Ω  →  φ  (  Ω  )  ⊆            R              n       is transformation of the coordinate systems and   D  φ  :  Ω  →            R              n      ×      n       is the corresponding Jacobi matrix.So in your case all you have to do is set   f  :≡  1 and write down your transformation   φ. Then       |    det  D  φ  (  x  )      |   is the expression you are looking for, and the above equation simplifies to   V  o  l  (  Ω  )  =      ∫          Ω        1  d  y  =      ∫                  φ                  −          1                    (      Ω      )            |    det  D  φ  (  x  )      |    d  x.

Finding volume of a sphere using integration.

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