Math 67 Chapter 14 Practice Exam
Name ______________________________
To receive full credit, I must be able to read and follow your work that supports your answer.
1. Find the Volume under the surface 𝑧 = 3𝑥√𝑥 2 + 𝑦 where 𝑅 = {(𝑥, 𝑦)|(0 ≤ 𝑥 ≤ 1, 0 ≤ 𝑦 ≤ 3}.
2. Find the Volume of the solid bounded by 𝑥 = 0; 𝑧 = 0;
𝑧 = 4 − 𝑥 2 ; 𝑦 = 3𝑥; 𝑦 = 9.
3. Use polar coordinates to evaluate ∬𝑅 2(𝑥 + 𝑦)𝑑𝐴 where 𝑅 is the region enclosed by
𝑥 2 + 𝑦 2 = 9; 𝑥 ≥ 0.
1
𝑧 √𝑦 2 +3
4. Evaluate ∫ ∫ ∫
−1 0 0
5. Evaluate
𝑦𝑧 𝑑𝑥 𝑑𝑦 𝑑𝑧.
√4−𝑦 2
√4−𝑥 2 −𝑦 2
2
∫−2 ∫−√4−𝑦 2 ∫−√4−𝑥 2 −𝑦 2
4√𝑥 2 + 𝑦 2 + 𝑧 2 𝑑𝑧 𝑑𝑥 𝑑𝑦 using
whichever coordinate system you prefer.
6. Sketch the region and evaluate the integral
𝜋
3
4
∫0 ∫0 ∫0 9 𝑟 2 𝑑𝑧 𝑑𝑟 𝑑𝜃 .
7. Use an appropriate transformation to find the area of the region in the first quadrant enclosed by
𝑥 + 𝑦 = 1; 𝑥 + 𝑦 = 2; 3𝑥 − 2𝑦 = 2; 3𝑥 − 2𝑦 = 5. Let 𝑢 = 𝑥 + 𝑦 𝑎𝑛𝑑 𝑣 = 3𝑥 − 2𝑦