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Monday August 18th, 2008 Name: MAT 21D - 021 Student ID#:
Practice Midterm Exam 1
MAT 21D - 021, Summer Session II 2008
Show your work on every problem. Correct answers with no supporting work will not receive full credit. Be organized and use notation appropriately. No calculators, notes, books, cellphones, etc. may be used on this exam. You should have only a pencil and eraser on your desk. Please write legibly!!
This is a PRACTICE midterm. It is by no means the only thing you should study. Use this as merely practice and nothing more!
Problem #1: Find the area of the region inside the circle r = 1 and outside the cardiod r = 1 − cos θ.
Problem #3: Write, but do not evaluate, a triple integral that computes the second moment about the z-axis of the unit sphere x^2 + y^2 + z^2 = 1, with constant density δ = 1.
Problem #4: Find the volume of the region bounded above by z = 4−x^2 −y^2 and below by z = 0.
Problem #6: Convert to cylindrical coordinates and evaluate: ∫ (^1)
− 1
∫ √ 1 −x 2
−√ 1 −x^2
∫ (^) (x (^2) +y (^2) )
−(x^2 +y^2 )
21 xy^2 dzdydx.
Problem #7: The surface of an apple can be described in spherical coordi- nates by the equation ρ = 1 − cos φ, a revolution figure of half a cardioid. Apples have constant density 1, find the mass of the apple.
EXTRA CREDIT
Problem #9: Evaluate: ∫ (^) π
0
∫ (^) π
0
∫ (^) π
0
cos(x + y + z) dxdydz.
