Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

y = 9 sin(x), y = 9 cos(x), 0 ≤ x ≤ /4; about y = −1

Answer 1

To find the volume of the solid obtained by rotating the region bounded by the curves y = 9 sin(x) and y = 9 cos(x), 0 ≤ x ≤ π/4, about the line y = -1, we can use the cylindrical shell method. This involves determining the height and radius of each cylindrical shell, calculating the volume of each shell, integrating the volume expression, and evaluating the integral to find the final volume.

Step-by-step explanation:

The volume V of the solid obtained by rotating the region bounded by the curves y = 9 sin(x) and y = 9 cos(x), 0 ≤ x ≤ π/4, about the line y = -1 can be found using the cylindrical shell method.

1. Determine the height of each cylindrical shell by subtracting the function values of y = 9 cos(x) and y = -1: (9cos(x) - (-1)) = 10 + 9cos(x).
2. Find the radius of each cylindrical shell by subtracting the function values of y = 9 sin(x) and y = -1: (9sin(x) - (-1)) = 10 + 9sin(x).
3. Calculate the volume of each cylindrical shell by multiplying the height and the circumference of the shell: 2π(10 + 9cos(x))(10 + 9sin(x)).
4. Integrate the volume expression from x = 0 to x = π/4: ∫(0 to π/4) 2π(10 + 9cos(x))(10 + 9sin(x)) dx.
5. Evaluate the integral to find the volume V.

Answer 2

Step-by-step explanation:

A cross-section is a washer with an inner radius of 8sin(x) - (-1) and an outer radius of 8cos(x) - -(1), so its area would be:

A(x) = π[(8cos(x) + 1)^2 − (8sin(x) + 1)^2]

= π[64cos^2(x) + 16cos(x) + 1 - 64sin^2(x) − 16sin(x) − 1]

= π[64cos(2x) + 16cos(x) - 16sin(x)]

=> V(x) = ∫[0,π/4] π[64cos(2x) + 16cos(x) - 16sin(x)] dx

= π[32sin(2x) + 16sin(x) + 16cos(x)] |[0,π/4]

= π[32sin(π/2) + 16√2/2 + 16√2/2 - 16]

= π(32 - 16 + 16√2) = π(16 + 16√2)

The volume of the region is π(16 + 16√2).