I am in Calculus and am currently learning how to find the Area of a Surface of Revolution. I cannot understand what the surface of revolution (whether it's the x-axis, y-axis, or y=6) is. For example, I had a problem saying to use the washer method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations y=x y=3 x=0 about the line y=4. I don't understand what I am supposed to do. Sometimes I can get the problem right with the way my teacher explained it and sometimes I can't. Please help!

First of all, you have to visualize what the surface of revolution is. You get it by rotating a line (straight or curved) 360 degrees about an axis, like the outside of a vase or pot being made on a lathe or potter's wheel.

Draw yourself a figure with the three lines plotted. In your case, the line that you are rotating is the straight horizontal line y=3 from x=0 to 3, and the 45 degree line y=x from x = 3 to 4.
When rotating the y = x line and y = 3 line about y=4, you get a pencil-shaped volume running from x=0 to x=4. You will have to do the integration in two parts: from x=0 to x=3, you get a cylinder with radius 1 and length 3 (from x=0 to x=3). The volume of that part is 3 pi. You don't even have to perform the integration to see that. The "washer method" consists of doing a dx integration of pi r^2 dx from 0 to 3, with r = 4-3 = 1. The region from x=3 to 4 is conical because the y=x curve forms the surface. The volume of that part is (1/3) pi
I get the complete volume to be
V = (10/3) pi.

I understand that finding the area of a surface of revolution can be challenging at first. Let's break down the problem you mentioned step by step and explain the concept along the way.

To find the volume of the solid generated by revolving the region bounded by the graphs, we need to use the washer method. This method involves summing up the volumes of infinitesimally thin washers or disks that make up the solid.

In this problem, you are given the region bounded by the graphs of y=x, y=3, and x=0. To visualize the region, you can sketch a graph of these equations. The region will be a triangle with vertices at (0, 0), (0, 3), and (3, 3).

Now, let's consider how this region is being revolved. The problem states that it is being revolved about the line y=4. This means that as we rotate the region, it forms a three-dimensional solid with a hole in the middle.

To find the volume of this solid, we need to consider the thickness of each washer or disk. We can imagine slicing the region into thin vertical strips parallel to the y-axis. As we rotate these vertical strips, they create washers.

Each washer has an inner radius (r) and an outer radius (R). In this case, the inner radius is the distance between the line of revolution (y=4) and the lower curve (y=x), and the outer radius is the distance between the line of revolution and the upper curve (y=3).

To find these radii, we need to express y=x in terms of x: x=y. Now, we can determine the relationship between the y-coordinate and the radii using the equation of the line of revolution (y=4). The inner radius (r) becomes 4 - x, and the outer radius (R) becomes 4 - 3 = 1.

Now that we have the radii, we can calculate the volume of each washer using the formula for the volume of a washer:

V = π * (R^2 - r^2) * dx

Here, dx represents the thickness of each washer, which corresponds to an infinitesimally small change in x. To find the total volume, we integrate this expression over the range of x that corresponds to the region we are revolving.

∫[x=0 to x=3] π * (1^2 - (4 - x)^2) dx

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

Remember to always sketch the region and visualize the rotation to better understand the problem. You can also try practicing similar problems to reinforce the concept of finding the volume of a surface of revolution using the washer method.

I hope this explanation helps you better understand how to approach this type of problem in Calculus!