Engineering Mathematics 233 Solutions: Double and triple integrals

for all (x, y) in the plane region bounded by the circle x 2 + y 2 + b 2 = a 2 ⇐⇒ x 2 + y 2 = a 2 − b 2 .(0, 0, a)Bz = bsphere x 2 + y 2 + z 2 = a 2The region B is best described in cylindrical coordinates, and this givesB : b ≤ z ≤ √ a 2 − r 20 ≤ r ≤ √ a 2 − b 20 ≤ θ ≤ 2π.Then,volume(B) ==== 2π= 2π∫ ∫ ∫1 dVB∫ 2π ∫ √ a 2 −b ∫ √ 2 a 2 −r 20 0∫ 2π ∫ √ a 2 −b 20 0∫ √ a 2 −b 2(= 2π − 1 3br dz dr dθ( √ a 2 − r 2 − b)r dr dθ√a2 − r 2 r − br dr0(√ )(a 2 − r 2 ) 3/2(3/2)(−2)− a2 −b 2br22 ∣0((b 2 ) 3/2 − (a 2 ) 3/2) − b 2 (a2 − b 2 )( a 3 − b 3) ( )= 2π − a2 b3 2 + b3 a3= 2π2 3 − a2 b2 + b3 .6)17. Sketch the region B whose volume is given by the triple integral∫ 4 ∫ (4−x)/2 ∫ (12−3x−6y)/4Rewrite the triple integral using the order of integration dV = dy dx dz.SolutionFrom the triple integral, the region B is described by0000 ≤ z ≤12 − 3x − 6y,4141 dz dy dx.