Question

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the curves y = x^3, y = 0, x = 1, and x = 4 about the y-axis.

Answer 1

To find the volume of the solid obtained by rotating the region bounded by the curves y = x^3, y = 0, x = 1, and x = 4 about the y-axis using the method of cylindrical shells, we can use the formula: V = 2π ∫[a,b] x * f(x) dx. Simplifying the integral and plugging in the limits of integration, we find that the volume is approximately 2.57π cubic units.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by the curves y = x^3, y = 0, x = 1, and x = 4 about the y-axis using the method of cylindrical shells, we can use the formula:

V = 2π ∫[a,b] x * f(x) dx

where [a,b] is the interval over which we are rotating the region, and f(x) is the function representing the curve y = x^3. In this case, [a,b] = [0,1] since we are rotating the region between x = 0 and x = 1.

1. First, we need to express x in terms of y. Since y = x^3, we can write x = y^(1/3).
2. Next, we need to find the limits of integration for y. Since the region is bounded by y = 0 and y = x^3, the limits of integration for y are [0,1].
3. Substituting x = y^(1/3) into the formula, we get:

V = 2π ∫[0,1] y^(1/3) * y dy

Simplifying this integral:

V = 2π ∫[0,1] y^(4/3) dy

Integrating, we get:

V = 2π * (3/7)y^(7/3) | [0,1]

Plugging in the limits of integration:

V = 2π * (3/7)(1)^(7/3) - 2π * (3/7)(0)^(7/3)

Simplifying:

V = 2π * (3/7)

V ≈ 2.57π