Compute the volume of the solid obtained by rotating the

Question

1. Use the Shell Method to compute the volume obtained by rotating the region enclosed by the graphs as indicated, about the y 1axis.

for y = 64 - x³, y = 64 - 16x, x ≥ 0

2. Use the Shell Method to compute the volume of the solid obtained by rotating the region underneath the graph of y = 1-x² over the interval [-1, 1] about the line x = 7.

3. Use the Shell Method to calculate the volume of rotation about the x - axis.

x = y(4 - y), x = 0

4. Let R be the region under the graph of y = 9 - x² for 0 ≤ x ≤ 2. Use the Shell Method to compute the volume of rotation of R about the x-axis as a sum of two integrals along the y -axis.

Hint: The shells generated depend on whether y ∈ [0, 5] or y ∈ [5, 9].

5. Use the Shell Method to find the volume V of the solid obtained by rotating the region above the graph of y = x² + 3 and below y = 7 for 0 ≤ x ≤ 2 about the line y = -3.

V =

6. Use the Shell Method to find the volume of the solid obtained by rotating the region below the graph of y = x² + 5 and above y = 0 for 0 ≤ x ≤ 5  about  x = 5.

V =