A sphere of radius R centered at a po P is the surface (in 3-dimensional space)

Question

A sphere of radius R centered at a po P is the surface (in 3-dimensional space) consisting of all points whose distance from the po P is R. In Cartesian coordinates, a sphere of radius R centered at P = (x0, y0, z0) is given implicitly by the equation (x - x0)2 + (y - y0)2 + (z - z0)2 = R2 Furthermore, we can parameterized a sphere of radius R centered at P by the mapping T : [0, 2pi] times [0, pi] rightarrow R3 defined by T(theta, phi) = (x = x0 + R sin phi cos theta, y = y0 + R sin phi sin theta, z = z0 + R cos phi). Plot a sphere of radius 1 centered at the origin using eq. (2) and the plot3d command. Use the options axes = boxed and view = -3..3 to better view the plot. Find a formula for the volume of the solid bounded by a sphere of radius R using anintegral. Use anintegral to determine a formula for the surface area of a sphere of radius R.

Explanation / Answer

1st integral using greens theorem

2nd integral using general method

3rd integral using the fundamental theorem

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