cive 221 2009 Quiz 4a - 2 x y z over the volume in the...

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CIVE 221 Fall 2009 Nov 20 th , 2009 1 of 1 Quiz #4 Name: ID#: Instructions: Quiz will be 50 minutes long. No programmable calculators. Marks: / 8 Q1 - (2 marks) Set up, but do not evaluate, a triple iterated integral with respect to z , y , and x for the triple integral of the function 2 x y z + over the volume in the first octant bounded by the surfaces z = 2- y , x 2 + y 2 = 4, x = 0, y = 0, z = 0. Step 1 – Draw V. Step 2 – contribution of a point: ( ) 2 x y z dzdydx + Step 3 – contribution of a column in the direction: ( ) 2 2 0 y dl x y z dzdydx = + Step 4 – contribution of all columns ( ) 2 2 0 y D D V dl x y z dzdydx = = + ∫∫ ∫∫∫ Step 5 – draw D
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CIVE 221 Fall 2009 Nov 20 th , 2009 2 of 1 Step 6 – Set up bounds of D ( ) 2 2 4 2 2 0 0 0 x y V x y z dzdydx = + ∫ ∫ ∫ Q2 - (2 marks) Using cylindrical coordinates, set up, but do not evaluate, a triple iterated integral with respect to z , r , and θ for the triple integral of the function
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Unformatted text preview: 2 x y z + over the volume in the first octant bounded by the surfaces z = y + 2, x 2 + y 2 = 4, x = 0, y = 0, z = 0. Step 1 – Draw V Step 2 – contribution of a point: ( ) ( ) 2 3 2 , , cos sin x y z dzdydx f x y z dzdydx r z rdzdrd θ + = = + Step 3 – contribution of a column in the direction: 2 sin 3 2 cos sin r dl r z rdzdrd + = + ∫ Step 4 – contribution of all columns CIVE 221 Fall 2009 Nov 20 th , 2009 3 of 1 ( ) ( ) 2 sin 2 2 sin 3 2 cos sin cos sin r D D r D V dl r r z rdzdrd V r z rdzdrd θ + + = = + = + ∫∫ ∫∫∫ ∫∫∫ Step 5 – draw D Step 6 – Set up bounds of D /2 2 2 sin 3 2 cos sin r V r z rdzdrd π + = + ∫ ∫∫...
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