Question

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Determine the volume of the solid that lies below x2 + y2 + (z-1/2)2 = 0.25 and above z = (x2 + y2)42

Answer 1

III GeoGebra 3D Calculator !!! SIGN IN 8 -1.6 1.4 14 -1.2 -0.8 -0.6 0.2 -0.2 0.4 0.6 - 1.2 A Activate Windows Go to Settings to activate Windows. 1.6

for spherical coordinates let x= psin & caso y = p sino sino = andy dz = gr sin odode dip za o cos where, s = dist from origin to the point P3. p=0 o = Yadar angle (as in polar coordinater); 0 50 52 a = angle from the zoones to say from origin through p. 3oEost so CS Scanned with CamScanner

2 Sol we want to find solution volume that lies between x² + y + (2-015)? 30125 & above (x² + y 2 / 2 Sketchis 91 YE sy XL ilz! I have attached a geogebra 3D blot for better understanding) use spherical coordinates to solve this i Let us Sphere XL +y? +: (Z-0.5)?= 0.25 +z? + 0,45 = 0.25 2 2 x - + Z sto 2 - £.cos Cone? Z = (x² + y 23% 18 = cosa => gcos d = (s² sin² d)/2 = f sind 6 =0,25TTT4 So, Required region is : s = cosd & d =TY4 sphere between Cone CS Scanned with CamScanner

Hence, 2T Volume = f * 4 caso P² sin & a g do do O o RT TT/4 - S sind (cos² d) do do 0 Let cos out ) sind d4 =-dt => udume s S-t at do o = 8 2 7 / 2T S-t de 1024 Tely 1. 26 11 f (cord)") de 0 up - 2 T [()-(03 # [ 1 - 4 = !! 3 cubic units 8 ty CS Scanned with CamScanner

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