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Volumes: Cylindrical Shells- HW Problems
Use the method of cylindrical shells to find the volume of the solid
obtained by rotating the region bounded by the curves about the given
line.
1. 𝑦 = 𝑥 − 𝑥 3 , 𝑦 = 0, 0 ≤ 𝑥 ≤ 1, about the 𝑦-axis
2. 𝑦 = 𝑥 3 , 𝑦 = 1, 𝑥 = 0, about the 𝑥-axis.
3. 𝑦 = 𝑥 2 + 2, 𝑦 = 0, 𝑥 = 0, 𝑥 = 2 about the 𝑦-axis.
4. 𝑦 = 𝑥 2 , 𝑦 = 𝑥, about the 𝑥-axis
5. 𝑦 = 𝑥 3 , 𝑦 = 1, 𝑥 = 0, about the line 𝑦 = 3
6. 𝑦 = 𝑥 2 , 𝑦 = 8 − 𝑥 2 , about the line 𝑥 = 2
7. 𝑦 = √9 − 𝑥, 𝑥 = 0, 𝑦 = 0, about the line 𝑦 = −2
8. 𝑦 = 4𝑥 − 𝑥 2 , 𝑦 = 3, about the line 𝑥 = 1.
Volumes: Cylindrical Shells- HW Problems
Use the method of cylindrical shells to find the volume of the solid
obtained by rotating the region bounded by the curves about the given
line.
1. 𝑦 = 𝑥 − 𝑥 3 , 𝑦 = 0, 0 ≤ 𝑥 ≤ 1, about the 𝑦-axis
2. 𝑦 = 𝑥 3 , 𝑦 = 1, 𝑥 = 0, about the 𝑥-axis.
3. 𝑦 = 𝑥 2 + 2, 𝑦 = 0, 𝑥 = 0, 𝑥 = 2 about the 𝑦-axis.
4. 𝑦 = 𝑥 2 , 𝑦 = 𝑥, about the 𝑥-axis
5. 𝑦 = 𝑥 3 , 𝑦 = 1, 𝑥 = 0, about the line 𝑦 = 3
6. 𝑦 = 𝑥 2 , 𝑦 = 8 − 𝑥 2 , about the line 𝑥 = 2
7. 𝑦 = √9 − 𝑥, 𝑥 = 0, 𝑦 = 0, about the line 𝑦 = −2
8. 𝑦 = 4𝑥 − 𝑥 2 , 𝑦 = 3, about the line 𝑥 = 1.
