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The Change of Variables Theorem- HW Problems
Evaluate the following integrals.
1. ∬𝐷 (𝑥 2 + 𝑦 2 )2 𝑑𝐴; where 𝐷 is the disk 𝑥 2 + 𝑦 2 ≤ 9.
2. ∬𝐷 (𝑥 2 + 𝑦 2 )2 𝑑𝐴; where 𝐷 is the annulus 1 ≤ 𝑥 2 + 𝑦 2 ≤ 4.
3. ∬𝐷 (𝑥 2 + 𝑦 2 )2 𝑑𝐴; where 𝐷 is the part of the annulus
1 ≤ 𝑥 2 + 𝑦 2 ≤ 4 where 𝑥 ≥ 0.
2 +𝑦 2 )
4. ∭𝑊 (𝑒 (𝑥 + 2𝑧)𝑑𝑉 where 𝑊 is part of the solid cylinder
𝑥 2 + 𝑦 2 ≤ 9 where 1 ≤ 𝑧 ≤ 2.
Hint: cylindrical coordinates.
5. ∭𝑊 (√𝑥 2 + 𝑦 2 + 𝑧 2 )𝑑𝑉 where 𝑊 is the set where
𝑥 2 + 𝑦 2 + 𝑧 2 ≤ 4.
Hint: spherical coordinates.
6. ∭𝑊 (√𝑥 2 + 𝑦 2 + 𝑧 2 )𝑑𝑉 where 𝑊 is the solid bounded by
𝑥 2 + 𝑦 2 + 𝑧 2 = 1, 𝑥 2 + 𝑦 2 + 𝑧 2 = 4,
and 𝑧 = 0, with 𝑧 ≥ 0.
The Change of Variables Theorem- HW Problems
Evaluate the following integrals.
1. ∬𝐷 (𝑥 2 + 𝑦 2 )2 𝑑𝐴; where 𝐷 is the disk 𝑥 2 + 𝑦 2 ≤ 9.
2. ∬𝐷 (𝑥 2 + 𝑦 2 )2 𝑑𝐴; where 𝐷 is the annulus 1 ≤ 𝑥 2 + 𝑦 2 ≤ 4.
3. ∬𝐷 (𝑥 2 + 𝑦 2 )2 𝑑𝐴; where 𝐷 is the part of the annulus
1 ≤ 𝑥 2 + 𝑦 2 ≤ 4 where 𝑥 ≥ 0.
2 +𝑦 2 )
4. ∭𝑊 (𝑒 (𝑥 + 2𝑧)𝑑𝑉 where 𝑊 is part of the solid cylinder
𝑥 2 + 𝑦 2 ≤ 9 where 1 ≤ 𝑧 ≤ 2.
Hint: cylindrical coordinates.
5. ∭𝑊 (√𝑥 2 + 𝑦 2 + 𝑧 2 )𝑑𝑉 where 𝑊 is the set where
𝑥 2 + 𝑦 2 + 𝑧 2 ≤ 4.
Hint: spherical coordinates.
6. ∭𝑊 (√𝑥 2 + 𝑦 2 + 𝑧 2 )𝑑𝑉 where 𝑊 is the solid bounded by
𝑥 2 + 𝑦 2 + 𝑧 2 = 1, 𝑥 2 + 𝑦 2 + 𝑧 2 = 4,
and 𝑧 = 0, with 𝑧 ≥ 0.
