[Mathematical Analysis for small readers] (volume and surface area of the area enclosed by 2014-10-27)

(From m.j. shu) sets up the body $ \ VSA $ by $ x ^ 2 + y ^ 2 = 2Z $ and $ Z = 4-\ SQRT {x ^ 2 + y ^ 2} $, evaluate the volume and surface area of $ \ VSA $.

 

A: The area is surrounded by a rotating parabolic table and a cone. the requested volume is $ \ beex \ Bea V & =\ int_0 ^ 2 \ pi \ cdot 2Z \ rd z + \ int_2 ^ 4 \ pi \ cdot (4-z) ^ 2 \ rd z \ & \ quad \ sex {x ^ 2 + y ^ 2 = 2Z, \ quad x ^ 2 + y ^ 2 = (4-z) ^ 2 }\\& = 4 \ PI + \ frac {8 \ PI} {3 }\\\& =\ frac {20 \ PI} {3 }. \ EEA \ eeex $ the requested surface area is $ \ beex \ Bea S & =\ iint _ {x ^ 2 + y ^ 2 \ Leq 4} \ sex {\ SQRT {1 + Z_1 '^ 2 + Z_2' ^ 2} + \ SQRT {1 + Z_2 '^ 2 + Z_2' ^ 2 }}\ RD x \ rd y \ & \ quad \ sex {Z_1 = \ frac {x ^ 2 + y ^ 2} {2 }, \ quad Z_2 = 4-\ SQRT {x ^ 2 + y ^ 2 }\\\&=\ iint _ {x ^ 2 + y ^ 2 \ Leq 4} \ SQRT {1 + x ^ 2 + y ^ 2} + \ SQRT {2} \ RD x \ rd y \ & =\ int_0 ^ 2 \ sex {\ SQRT {1 + R ^ 2 }+ \ SQRT {2 }}\ cdot 2 \ pi r \ rd r \\\&=\ frac {2} {3} \ sex {5 \ SQRT {5} + 6 \ SQRT {2}-1} \ pi. \ EEA \ eeex $

[Mathematical Analysis for small readers] (volume and surface area of the area enclosed by 2014-10-27)