Questions: What is the volume of the solid of revolution obtained by rotating the region bounded by y=2, y=x^3+1, and the y-axis around the y-axis? Select one: a. 3π/5 b. 2π/5 c. 2π/3 d. 2π/15

Solution Steps

To find the volume of the solid of revolution obtained by rotating the region bounded by \( y = 2 \), \( y = x^3 + 1 \), and the \( y \)-axis around the \( y \)-axis, we can use the method of cylindrical shells. The volume \( V \) is given by the integral:

\[ V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx \]

where \( f(x) \) is the height of the shell and \( x \) is the radius. Here, \( f(x) = 2 - (x^3 + 1) \) and the bounds are determined by the intersection points of \( y = 2 \) and \( y = x^3 + 1 \).

1. Determine the bounds of integration by solving \( 2 = x^3 + 1 \).
2. Set up the integral for the volume using the method of cylindrical shells.
3. Evaluate the integral to find the volume.

We need to find the volume of the solid of revolution obtained by rotating the region bounded by \( y = 2 \), \( y = x^3 + 1 \), and the \( y \)-axis around the \( y \)-axis.

The region is bounded by \( y = 2 \) and \( y = x^3 + 1 \). To find the bounds for \( x \), we solve for \( x \) when \( y = 2 \): \[ 2 = x^3 + 1 \implies x^3 = 1 \implies x = 1 \] Thus, the region is bounded by \( x = 0 \) (the \( y \)-axis) and \( x = 1 \).

We use the method of cylindrical shells to find the volume. The formula for the volume of a solid of revolution around the \( y \)-axis is: \[ V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx \] Here, \( f(x) = x^3 + 1 \), \( a = 0 \), and \( b = 1 \).

Set up the integral: \[ V = 2\pi \int_{0}^{1} x (x^3 + 1) \, dx \] Simplify the integrand: \[ V = 2\pi \int_{0}^{1} (x^4 + x) \, dx \] Integrate term by term: \[ V = 2\pi \left[ \frac{x^5}{5} + \frac{x^2}{2} \right]_{0}^{1} \] Evaluate the definite integral: \[ V = 2\pi \left( \left. \frac{x^5}{5} + \frac{x^2}{2} \right|_{0}^{1} \right) \] \[ V = 2\pi \left( \frac{1^5}{5} + \frac{1^2}{2} - \left( \frac{0^5}{5} + \frac{0^2}{2} \right) \right) \] \[ V = 2\pi \left( \frac{1}{5} + \frac{1}{2} \right) \] \[ V = 2\pi \left( \frac{2}{10} + \frac{5}{10} \right) \] \[ V = 2\pi \left( \frac{7}{10} \right) \] \[ V = \frac{14\pi}{10} \] \[ V = \frac{7\pi}{5} \]

Final Answer