Use a Triple integral to find the volume of the indicated region. The region enc

Question

Use a Triple integral to find the volume of the indicated region. The region enclosed by the paraboloids z=x^2+y^2-4 and z=28-x^2-y^2. whole and detailed answer, thanks

Explanation / Answer

example Use a triple integral and an appropriate change of coordinates to calculate the volume of the region bounded by the xy-plane, the surfacez = x^2 + y^2 and the cylinder over (x^2)/9 + (y^2)/4= 1. Use elliptical ('modified' polar) coordinates x = 3r cos ?, y = 2r sin ?, z = z. So, z = x^2 + y^2 ==> z = 9r^2 cos^2(?) + 4r^2 sin^2(?). and x^2 / 9 + y^2 / 4 = 1 ==> r = 1, the unit circle. Moreover, the Jacobian ?(x,y,z)/?(r,?,z) equals |3 cos ?....2 sin ?....0| |-3r sin ?..2r cos ?...0| = 6r. |....0............0........1| Hence, the volume ??? 1 dV equals ?(? = 0 to 2p) ?(r = 0 to 1) ?(z = 0 to 9r^2 cos^2(?) + 4r^2 sin^2(?)) 1 * (6r dz dr d?) = ?(? = 0 to 2p) ?(r = 0 to 1) 6r (9r^2 cos^2(?) + 4r^2 sin^2(?)) dr d? = ?(? = 0 to 2p) (9 cos^2(?) + 4 sin^2(?)) d? * ?(r = 0 to 1) 6r^3 dr = ?(? = 0 to 2p) (5 cos^2(?) + 4) d? * [(3/2)r^4 {for r = 0 to 1}] = (3/2) * ?(? = 0 to 2p) (5 * (1/2)(1 + cos(2?)) + 4) d? = (3/4) * ?(? = 0 to 2p) (5(1 + cos(2?)) + 8) d? = (3/4) * ?(? = 0 to 2p) (13 + 5 cos(2?)) d? = (3/4) * (13 * 2p + 0) = 39p/2.

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