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Triple Integrals- HW Problems
Evaluate the following integrals.
1. ∭𝐵 (2𝑥 + 4𝑦 + 6𝑧)𝑑𝑉, where 𝐵 = [0,1] × [0,1] × [0,1].
2. ∭𝐵 (𝑦𝑒 (𝑥+𝑧) )𝑑𝑉, where 𝐵 = [0,1] × [0,1] × [0,2].
3. Write a triple integral with endpoints, but do not evaluate, that
represents the volume of the solid bounded by the cylinder
𝑥 2 + 𝑦 2 = 4, and the planes 𝑧 = 0, 2𝑥 + 𝑦 + 𝑧 = 7 (Draw the solid).
Evaluate the following integrals.
4. ∭𝑊 (3𝑧 2 )𝑑𝑉 where 𝑊 is the solid bounded by the planes
𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑧 = 1, and the cylinder 𝑥 2 + 𝑦 2 = 1, where
𝑥 ≥ 0, 𝑦 ≥ 0 (Draw 𝑊).
2 𝑥 𝑥+𝑦 4
5. ∫0 ∫0 ∫0 (𝑒 (𝑥 ) (2𝑦 + 2𝑧)) 𝑑𝑧𝑑𝑦𝑑𝑥.
6. Using a triple integral, find the volume of the solid bounded by
𝑧 = 9 − 𝑥 2 − 𝑦 2 and the 𝑥-𝑦 plane.
7. Set up with limits of integration, but do not evaluate, a triple
integral to find the volume of the solid bounded by
𝑥 2 + 𝑦 2 + 𝑧 2 = 1, 𝑥 2 + 𝑦 2 + 𝑧 2 = 9, 𝑧 = 0, where 𝑥, 𝑦, 𝑧 ≥ 0.
Triple Integrals- HW Problems
Evaluate the following integrals.
1. ∭𝐵 (2𝑥 + 4𝑦 + 6𝑧)𝑑𝑉, where 𝐵 = [0,1] × [0,1] × [0,1].
2. ∭𝐵 (𝑦𝑒 (𝑥+𝑧) )𝑑𝑉, where 𝐵 = [0,1] × [0,1] × [0,2].
3. Write a triple integral with endpoints, but do not evaluate, that
represents the volume of the solid bounded by the cylinder
𝑥 2 + 𝑦 2 = 4, and the planes 𝑧 = 0, 2𝑥 + 𝑦 + 𝑧 = 7 (Draw the solid).
Evaluate the following integrals.
4. ∭𝑊 (3𝑧 2 )𝑑𝑉 where 𝑊 is the solid bounded by the planes
𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑧 = 1, and the cylinder 𝑥 2 + 𝑦 2 = 1, where
𝑥 ≥ 0, 𝑦 ≥ 0 (Draw 𝑊).
2 𝑥 𝑥+𝑦 4
5. ∫0 ∫0 ∫0 (𝑒 (𝑥 ) (2𝑦 + 2𝑧)) 𝑑𝑧𝑑𝑦𝑑𝑥.
6. Using a triple integral, find the volume of the solid bounded by
𝑧 = 9 − 𝑥 2 − 𝑦 2 and the 𝑥-𝑦 plane.
7. Set up with limits of integration, but do not evaluate, a triple
integral to find the volume of the solid bounded by
𝑥 2 + 𝑦 2 + 𝑧 2 = 1, 𝑥 2 + 𝑦 2 + 𝑧 2 = 9, 𝑧 = 0, where 𝑥, 𝑦, 𝑧 ≥ 0.
