Find the volume of the solid whose base is the region bounded by the graphs of y=x^3,x=1, and the x-axis, and whose cross sections perpenditular to the x-axis are semicircles.

What would be the radius in this case?
I thought it would just be x^3, but others are saying (x^3)/2. Please help me!

If you rotate around y=0, the x axis, the outer radius is x^3, but note the rotation is only half a circle. If you do a rotational volume based on average radius, it is x^3/2. I would never do it that way, but some do.

There are a number of ways to do this.

I don't quite understand what you mean by "based on average radius"

Dont do it that way if you don't understand centroids.

I apologize for the confusion. When I mentioned "based on average radius," I was referring to a method called the Method of Average Radius, which is an alternative method to find the volume in certain cases.

In this specific problem, the cross sections are semicircles. To find the volume of the solid, you can use either the Method of Discs (or Disks) or the Method of Cylindrical Shells.

Let's focus on the Method of Discs for now. For each point x in the interval [0, 1], the cross section is a semicircle with a diameter parallel to the x-axis. Since the y-coordinate of each point in the curve y = x^3 is equal to the radius of the corresponding semicircle, you are correct that the radius of each semicircle is x^3.

To find the volume using the Method of Discs, you integrate the areas of the cross sections over the interval [0, 1]. The formula for the area of a circle is A = πr^2, where r is the radius. However, since we have semicircles, we only take half of the area, so the formula becomes A = (1/2)πr^2.

Substituting the known radius r = x^3, we have A = (1/2)π(x^3)^2 = (1/2)πx^6. Integrating this expression over the interval [0, 1] gives the volume of the solid.

V = ∫[0, 1] (1/2)πx^6 dx

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

As for the claim that (x^3)/2 is the average radius, it seems to refer to the Method of Average Radius. This method involves averaging the inner and outer radii of the cross sections when rotating a region to find the volume. However, it is not applicable in this case because the cross sections are semicircles, not full circles.

I hope this explanation clarifies things for you. Let me know if you have any further questions!