Determine the solid's volume inside each of the two spheresx2+y2+z2+4x−2y+4z+5=0&nbsp;and&nbsp;x2+y2+z2=4

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Determine the solid&#039;s volume inside each of the two spheresx2+y2+z2+4x−2y+4z+5=0&amp;nbsp;and&amp;nbsp;x2+y2+z2=4

wornoutwomanC · Accepted Answer

Step1&amp;nbsp;x2+y2+z2+4x−2y+4z+5=0&amp;nbsp;(x2+4x)+(y2−2y)+(z2+4z)+5=0&amp;nbsp;(x2+4x+4−4)+(y2−2y+1−1)+(z2+4z+4−4)+5=0&amp;nbsp;(x2+4x+4)−4+(y2−2y+1)−1+(z2+4z+4)−4+5=0&amp;nbsp;(x+2)2+(y−1)2+(z+2)2=4&amp;nbsp;This sphere has a radius of 2 and a center at (-2,1,-2)Step 2&amp;nbsp;I am going to use the following formula :&amp;nbsp;V=π12(4R+d)(2R−d)2&amp;nbsp;where d is the separation between the centers of two spheresAnd R is the radius of the two spheres, that is 2 in this case&amp;nbsp;Link to the proof of this formula is given in comment section.&amp;nbsp;Step 3&amp;nbsp;Distance between the two centres is&amp;nbsp;d=(−2)2+1(1)2+(−2)2=4+1+4=9=3&amp;nbsp;Step 4&amp;nbsp;Substitute d=3 and R=2, to get the volume&amp;nbsp;V=π12(4⋅2+3)(2⋅2−3)2≈2.88&amp;nbsp;Result&amp;nbsp;V≈2.88

Find the volume of the solid that lies inside both of the spheres x^2+y^2

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