Consider the solid that is bounded below by the cone z=3x2+3y2 and above by the sphere x2+y2+z2=16..Set up only the appropriate triple integrals in cylindrical and spherical coordinates needed to find the volume of the solid.

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Consider the solid that is bounded below by the cone z=3x2+3y2  and above by the sphere x2+y2+z2=16..Set up only the appropriate triple integrals in cylindrical and spherical coordinates needed to find the volume of the solid.

averes8 · Accepted Answer

Step 1So we need, Set up only the appropriate triple integrals in cylindrical and spherical coordinates needed to find the volume of the solid.To set the triple integral in cylindrical coordinatesBy using relation,x=rcos⁡θy=rsin⁡θz=zThus,The cone&amp;nbsp;z=3x2+3y2&amp;nbsp;in cylindrical coordinate becomes,z=3r2=3rAnd the sphere become&amp;nbsp;r2+z2=16To find the limit of r,Consider,⇒3r2=16−r2⇒3r2=16−r2⇒r2=4⇒r=2Step 2Thus, we can describe the region asE={(r,θ,z)|0≤θ≤2π|0≤r≤23r≤z≤16−r2}Hence, the triple integral for the volume by cylindrical coordinates isV=∫02π∫02∫3r16−r2rdzdrdθStep 3Now, to set the triple integral in spherical coordinatesSince,p2=x2+y2+z2tan⁡θ=yxφ=arccos⁡(zx2+y2+z2)The sphere&amp;nbsp;x2+y2+z2=16&amp;nbsp;gives&amp;nbsp;p2=16=&amp;gt;p=4From the conez=3x2+3y2=3r⇒pcos⁡(φ)=3psin⁡(φ)⇒ptan⁡(φ)=13φ=π6Step 4Thus, we can describe the region asE={(p,φ,θ)|0≤p≤4,0≤φ≤π6,0≤θ≤2π}Hence, the triple integral for the volume of the solid by spherical coordinate isV=∫02π∫0π6∫04p2sin⁡(φ)&amp;nbsp;d&amp;nbsp;pd&amp;nbsp;φ&amp;nbsp;d&amp;nbsp;θStep 5Now, evaluating the integral of cylindrical coordinate we get.⇒V=∫02π∫02∫3r16−r2rdzdrdθ⇒V=17.9582And evaluating the integral of spherical coordinate⇒V=∫02π∫0π6∫04r2sin⁡(φ)dpdφdθ⇒V=17.9582Thus, by both coordinate systems, we get the same volume.Therefore, both triple integrals are appropriate.

Consider the solid that is bounded below by the cone z = sqrt{3x^{2}+3y^{2}} and above by the sphere x^{2} +y^{2} + z^{2} = 16..Set up only the appropriate triple integrals in cylindrical and spherical coordinates needed to find the volume of the solid.

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