Midterm Exam 3 Solution Summer 2014 on Multivariable Calculus

# Midterm Exam 3 Solution Summer 2014 on Multivariable Calculus

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Math 2263 Exam 3 - Page 2 of 7 7/25/14 1. Let E be the solid region above the plane z = 2 and below the upper hemisphere of the sphere x 2 + y 2 + z 2 = 16. Set up (but do not evaluate ) an iterated integral that would calculate the triple integral ZZZ E ( y + z ) dV using: (a) (10 points) Cylindrical coordinates. Solution: In the order dzdrdθ , the integral is: Z 2 π 0 Z 12 0 Z 16 - r 2 2 ( r sin θ + z ) r dzdrdθ The upper limit of 16 - r 2 for z is coming from the equation of the upper hemisphere written in cylindrical coordinates. The upper limit of 12 for r is coming from the fact that the plane z = 2 intersects the sphere in a circle with radius
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