Find the volume of the solid bounded by the graphs of these equations: z = 10-x2

Question

Find the volume of the solid bounded by the graphs of these equations: z = 10-x2-y2, y=x2,x = y2, and z = 0. Find the centroid of the solid in the first octant bounded by the planes y = 0 and z = 0 and by the surfaces z = 4 - x2 and x = y2. Find the volume of the solid right cylinder whose base is the region in the xy - plane that lies inside the cardioid r = 1 + cos theta and outside the circle r = 1 and whose top lies in the plane z = 4. Find the volume of the smaller region cut from the s^lid sphere rho 2 by the plane z = 1. Find the volume of the region between the spheres x2 + y2 + z2=1 and x2 + y2 + z2 = 4.

Explanation / Answer

4.

In spherical coordinates, z = ? cos ? = 1 ==> ? = 1/cos ?.
Intersecting this with the sphere yields 1/cos ? = 2 ==> cos ? = 1/2 ==> ? = ?/3.

Hence, the volume ??? 1 dV equals
?(? = 0 to 2?) ?(? = 0 to ?/3) ?(? = 1/cos ? to 2) 1 * (?^2 sin ? d? d? d?)
= 2? ?(? = 0 to ?/3) (1/3) ?^3 sin ? {for ? = 1/cos ? to 2} d?
= (2?/3) ?(? = 0 to ?/3) (2^3 - (1/cos ?)^3) sin ? d?
= (2?/3) ?(? = 0 to ?/3) [8sin ? - tan ? sec^2(?)] d?
= (2?/3) [-8cos ? - tan^2(?)] {for ? = 0 to ?/3}

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