192FinalSpring07

# 192FinalSpring07 - Math 192[5 Final Exam 1 Spring 2007 z...

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Math 192 Final Exam Spring 2007 1 z y x [5] 1. (a) Write down a triple integral for the volume of the solid in the ﬁrst octant bounded by the planes x + y + z = 2 and z =1. [5] (b) Compute the volume using the triple integral. [10] 2. Evaluate the integral Z 1 0 Z 1 y 1 / 3 1 x 4 +1 dx dy . [12] 3. Evaluate the integral Z 1 0 Z 1 - x 2 0 Z 1+ 1 - x 2 - y 2 ( x 2 + y 2 ) 1 / 4 2 z tan - 1 ( y/x ) dz dy dx . Bonus: Sketch the region of integration. [5] 4. (a) Consider a rectangular solid resting on the xy -plane with faces parallel to the coordinate planes and with its upper corners on the sphere x 2 + y 2 + z 2 = 1. If the corner in the ﬁrst octant is at the point ( a, b, c ) and the solid is made from a material of variable density δ ( x, y, z )= z , show that the mass of the solid is M ( a, b, c )=2 abc 2 . [10] (b) Find the maximum mass that the solid in part (a) can have. [10] 5. Find the points on the ellipsoid x 2 +4 y 2 +4 z 2 = 21 where the tangent plane is parallel to the plane 2

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192FinalSpring07 - Math 192[5 Final Exam 1 Spring 2007 z...

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