Question

Find the volume V of the solid obtained by rotating the region

bounded by the given curves about the specified line. y = x3, y =
1, x = 2; about y = −2

Answer 1

The volume V of the solid obtained by rotating the region bounded by the curves
`y = x^3` , y = 1 , and x = 2 about the line y = -2 is approximately 91.34 cubic units.

Step-by-step explanation:

To find the volume of the solid, we use the disk method, which involves integrating the cross-sectional areas of infinitesimally thin disks perpendicular to the axis of rotation. In this case, the axis of rotation is y = -2. The limits of integration are determined by the points of intersection of the curves
`\( y = x^3 \), \( y = 1 \), and \( x = 2 \)`.

The volume V is given by the integral
`\(\pi \int_(0)^(2) [(f(x))^2 - (g(x))^2] \,dx\), where \( f(x) \) and \( g(x) \)` are the upper and lower functions, respectively. In this scenario,
`\( f(x) = x^3 \) and \( g(x) = -2 \)`. Thus, the integral becomes
`\(\pi \int_(0)^(2) [(x^3 + 2)^2 - (-2)^2] \,dx\)`. Evaluating this integral yields the final volume V of approximately 91.34 cubic units. The negative sign of the lower limit accounts for the rotation about the line y = -2 .

In summary, the disk method provides an effective approach for calculating volumes of solids obtained by rotating regions bounded by curves. The choice of upper and lower functions, along with appropriate limits, is crucial for accurate calculations.

Answer 2

To find the volume of the solid obtained by rotating the region bounded by the curves about the line y = -2, we can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by the curves about the line y = -2, we can use the method of cylindrical shells.

We need to find the limits of integration, the height of the shell, and the radius of the shell at a given height.

The region bounded by the curves is a rectangle with base 2 units long and height 1. The height of the shell varies from -2 to 1. The radius of the shell at a given height is the difference between the x-value of the curve y = x^3 and the x-value of the curve y = 1.

The volume of each shell is given by the formula V = 2πrh, where r is the radius of the shell and h is the height of the shell.

Using the limits of integration (-2 to 1), we can set up the integral for the volume as follows:

V = ∫[from -2 to 1] 2π[(x^3 - 2)^2] dx.