*Note I reposted this question as I changed the subject**


The Region "R" under the graph of y = x^3 from x=0 to x=2 is rotated about the y-axis to form a solid.

a. Find the area of R.
b. Find the volume of the solid using vertical slices.
c. Find the first moment of area of R with respect to the y-axis. What do you notice about the integral?
d. Find the x coordinate of the centroid of R.
e. A theorem of Pappus states that the volume of a solid of revolution equals the area of the region being rotated times the distance the centroid of the region travels. Show that this problem confirms this theorem.

I was able to do part "a" as the integral from 0 to 2 of x^3 dx. Also I believe part "b" is pi*[3y^(5/3)/5] evaluated from 0 to 2

Too advanced for you?

To solve the remaining parts of the problem, let's go step by step:

b. To find the volume of the solid using vertical slices, we need to integrate the area of each slice over the range of y values. Since we are rotating the region around the y-axis, the slices are perpendicular to the y-axis.

The area of each slice can be calculated as the cross-sectional area of a disk, which is given by A = πr^2, where r is the distance from the y-axis to the edge of the slice. In this case, r is equal to the x-coordinate of the function y = x^3.

So, the volume of the solid can be found by integrating the area of each disk from 0 to 2:

V = ∫(0 to 2) π(x^3)^2 dy

Simplifying that expression, we get:

V = ∫(0 to 2) πx^6 dy

To evaluate this integral, we need to express x in terms of y. Since y = x^3, we can solve for x to get x = y^(1/3). Now, we can rewrite the integral:

V = ∫(0 to 2) π(y^(1/3))^6 dy

V = ∫(0 to 2) πy^(2) dy

Evaluating this integral from 0 to 2 will give you the volume of the solid.

c. The first moment of area of a region with respect to an axis is given by the integral of the product of the area element and its distance from the axis. For the region R rotating around the y-axis, the distance from the axis to a given point is simply x.

So, the moment of area with respect to the y-axis can be found by integrating x * dA, where dA represents an infinitesimally small area element.

Let's express x as a function of y, using x = y^(1/3):

∫(0 to 2) x * dA = ∫(0 to 2) (y^(1/3)) * dA

The moment of area integral is related to the first moment by multiplying the known area of the region:

∫(0 to 2) x * dA = A * ∫(0 to 2) (y^(1/3)) * dy

We already calculated A in part a.

d. To find the x-coordinate of the centroid of R, we need to evaluate the integral of x * dA, divided by the total area of R.

Centroid_x = (1/Area) * ∫(0 to 2) x * dA

Use the A value obtained in part a for the area.

e. To show that the problem confirms Pappus' Theorem, we need to compare the calculated volume using vertical slices (part b) with the product of the area of the region and the distance traveled by the centroid.

According to Pappus' Theorem, Volume = Area * Distance

Calculate the product of the area of R, obtained in part a, and the distance traveled by the centroid, obtained in part d. Compare this with the volume calculated in part b.

If the two values are equal, then it confirms Pappus' Theorem for this problem.

I hope this helps you solve the remaining parts of the problem! Let me know if you have any further questions.