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Applications of Multiple Integrals- HW Problems
2 +𝑦2 )
1. Find the average value of the function 𝑓(𝑥, 𝑦) = 𝑒 −(𝑥 over the
region where 4 ≤ 𝑥 2 + 𝑦 2 ≤ 9 and 𝑥 ≤ 0.
2. Find the center of mass of a uniformly dense region in ℝ2 bounded
by 𝑦 = √4 − 𝑥 2 , 𝑦 = 0, and 𝑥 = 0, with 𝑥 ≥ 0.
3. Find the mass of a solid bounded by the cylinder 𝑥 2 + 𝑦 2 = 4 and
the cone 𝑧 2 = 𝑥 2 + 𝑦 2 if the density is given by 𝛿 (𝑥, 𝑦, 𝑧) = √𝑥 2 + 𝑦 2 .
4. Set up the integrals with endpoints, but do not evaluate, that
represent the center of mass coordinates of a solid region bounded by
𝑥 2 + 𝑦 2 + 𝑧 2 = 1 and 𝑥 2 + 𝑦 2 + 𝑧 2 = 4, with 𝑧 ≥ 0 and
𝛿 (𝑥, 𝑦, 𝑧) = 𝑥 2 + 𝑦 2 + 𝑧 2 .
5. Find the moment of inertia about the 𝑧 axis of a ball given by
𝑥 2 + 𝑦 2 + 𝑧 2 ≤ 4 if the density is constant.
Applications of Multiple Integrals- HW Problems
2 +𝑦2 )
1. Find the average value of the function 𝑓(𝑥, 𝑦) = 𝑒 −(𝑥 over the
region where 4 ≤ 𝑥 2 + 𝑦 2 ≤ 9 and 𝑥 ≤ 0.
2. Find the center of mass of a uniformly dense region in ℝ2 bounded
by 𝑦 = √4 − 𝑥 2 , 𝑦 = 0, and 𝑥 = 0, with 𝑥 ≥ 0.
3. Find the mass of a solid bounded by the cylinder 𝑥 2 + 𝑦 2 = 4 and
the cone 𝑧 2 = 𝑥 2 + 𝑦 2 if the density is given by 𝛿 (𝑥, 𝑦, 𝑧) = √𝑥 2 + 𝑦 2 .
4. Set up the integrals with endpoints, but do not evaluate, that
represent the center of mass coordinates of a solid region bounded by
𝑥 2 + 𝑦 2 + 𝑧 2 = 1 and 𝑥 2 + 𝑦 2 + 𝑧 2 = 4, with 𝑧 ≥ 0 and
𝛿 (𝑥, 𝑦, 𝑧) = 𝑥 2 + 𝑦 2 + 𝑧 2 .
5. Find the moment of inertia about the 𝑧 axis of a ball given by
𝑥 2 + 𝑦 2 + 𝑧 2 ≤ 4 if the density is constant.
