Change from rectangular to cylindrical coordinates. (Let r≥0 and 0≤θ≤2π.) a) (−2,2,2) b) (−9,93,6) c) Use cylindrical coordinates. Evaluate ∫∫∫ExdV where E is enclosed by the planes z=0 and z=x+y+10 and by the cylinders x2+y2=16 and x2+y2=36 d) Use cylindrical coordinates. Find the volume of the solid that is enclosed by the cone z=x2+y2 and the sphere x2+y2+z2=8.

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Change from rectangular to cylindrical coordinates. (Let r≥0 and 0≤θ≤2π.)  a) (−2,2,2)  b) (−9,93,6)  c) Use cylindrical coordinates.  Evaluate  ∫∫∫ExdV  where E is enclosed by the planes z=0 and  z=x+y+10  and by the cylinders  x2+y2=16 and x2+y2=36  d) Use cylindrical coordinates.  Find the volume of the solid that is enclosed by the cone  z=x2+y2  and the sphere  x2+y2+z2=8.

wornoutwomanC · Accepted Answer

Step 1&amp;nbsp;Convert coordinates from rectangular to cylindrical.Let&amp;nbsp;r≥0&amp;nbsp;and&amp;nbsp;0≤θ≤2π&amp;nbsp;a)&amp;nbsp;(x,y,z)=(−2,2,2)&amp;nbsp;Use cylindrical coordinates.&amp;nbsp;r=x2+y2=(−2)3+23=22&amp;nbsp;θ=arctan⁡(yx)=arctan⁡(2−2)(−1)=3π4&amp;nbsp;z=z=2&amp;nbsp;Consequently, the necessary coordinates are,(r,θ,z)=(22,2π4,2)&amp;nbsp;Step 2&amp;nbsp;b)&amp;nbsp;(x,y,z)=(−9.93.6)&amp;nbsp;Use cylindrical coordinates.&amp;nbsp;r=x2+y2=(−9)2+(93)2=18&amp;nbsp;θ=arctan⁡(yx)=arctan⁡(93−9)=arctan⁡(−3)=2π3&amp;nbsp;z=z=6&amp;nbsp;Therefore, the required coordinates is,&amp;nbsp;(r,θ,z)=(18,2π3,6)&amp;nbsp;Step 3&amp;nbsp;c) Use cylindrical coordinates, to evaluate&amp;nbsp;∫∫E∫xdV&amp;nbsp;Where E is enclosed by the planes&amp;nbsp;z=0&amp;nbsp;and&amp;nbsp;z=x+y+10&amp;nbsp;and by the cylinders&amp;nbsp;x2+y2=16&amp;nbsp;and&amp;nbsp;x2+y2=36&amp;nbsp;16≤x2+y2≤36&amp;nbsp;⇒16≤r2≤36&amp;nbsp;As&amp;nbsp;x2+y2=r2&amp;nbsp;⇒4≤r≤6&amp;nbsp;and&amp;nbsp;0≤θ≤2π&amp;nbsp;and&amp;nbsp;0≤z≤x+y+10&amp;nbsp;⇒0≤z≤rcos⁡θ+rsin⁡θ+10&amp;nbsp;As&amp;nbsp;x=rcos⁡θ,y=rsin⁡θ&amp;nbsp;and&amp;nbsp;dV=dxdydz=rdrdθdz&amp;nbsp;So,&amp;nbsp;∫∫E∫xdV=∫02a∫46∫0rcos⁡θ+rsin⁡θ+10rcos⁡θ(rdrdθdz)&amp;nbsp;=∫02π∫46∫0rcos⁡θ+rsin⁡θ+10r2cos⁡θdrdθdz&amp;nbsp;=∫02π∫46r2(rcos⁡θ+rsin⁡θ+10)cos⁡θdrdθ&amp;nbsp;=∫02π∫46(r3(cos2⁡θ+sin⁡θcos⁡θ)+10r2cos⁡θ)drdθ&amp;nbsp;&amp;nbsp;

Change from rectangular to cylindrical coordinates. (Let r\geq0 and 0\leq\theta\leq2\pi.) a) (-2, 2, 2) b) (-9,9\sqrt{3,6}) c) Use cylindrical coordin

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