Which of the following represents the volume of the solid formed by revolving the region bounded by the graphs of y =x^3, y = 1, and x = 2, about the line x = 2?

I meant to phrase the question as, "What integral represents the volume of the solid formed by revolving the region bounded by the graphs of y =x^3, y = 1, and x = 2, about the line x = 2?"

Well, let's gather the clowndidates and calculate the volume of this solid. To find the volume, we can use the Disk Method since we're revolving the region around a vertical line, which in this case is x = 2. So, grab your circus hats and let's get rolling!

First, we need to determine the limits of integration. The region bounded by the graphs of y = x^3, y = 1, and x = 2 is bounded by x = 0 and x = 2 because that's where the boundaries intersect.

Next, we need to express the radius of each disk. Since we're revolving around x = 2, the radius of each disk is simply the distance between x = 2 and the curve. In this case, the distance is 2 - x.

Now, let's form the integral to calculate the volume:

V = ∫[from 0 to 2] π(2 - x)^2 dx

Integrating this expression will give us the volume of the solid formed by revolving the region bounded by the graphs. Plug it into your favorite calculator and you'll be spinning with the answer in no time!

To find the volume of the solid formed by revolving the region bounded by the given graphs, we can use the method of cylindrical shells.

First, we need to determine the limits of integration. In this case, we are revolving the region bounded by the graphs of y = x^3, y = 1, and x = 2. The region lies between x = 0 and x = 2.

Next, we consider an infinitesimally thin vertical strip at a distance x from the line x = 2. This strip has a width dx and a height equal to the difference in the y-values between the two curves at that point, which is: (x^3 - 1). The radius of the cylindrical shell is the distance from the line x = 2 to the x-coordinate of the strip, which is (2 - x).

The volume of each cylindrical shell can be calculated as follows:
dV = 2πr * h * dx
= 2π(2 - x)(x^3 - 1) dx

To find the total volume, we integrate this expression over the interval [0, 2]:
V = ∫[0,2] 2π(2 - x)(x^3 - 1) dx

Evaluating this integral will give us the volume of the solid.

By integrating the expression 2π(2 - x)(x^3 - 1) with respect to x over the interval [0, 2] using either manual methods or software, we would obtain the value representing the volume of the solid formed by revolving the given region about the line x = 2.

using shells of thickness dx,

v = ∫[1,2] 2π(2-x)(x^3-1) dx

using discs of thickness dy,

v = ∫[1,8] π(2-∛y)^2 dy