How do you find the volume of a solid?

Example 3 Determine the volume of the solid obtained by rotating the region bounded by y = x2 −2x y = x 2 − 2 x and y = x y = x about the line y = 4 y = 4 . First let’s get the bounding region and the solid graphed.

How do you find the surface of a solid of revolution?

To get a solid of revolution we start out with a function, y = f (x) y = f (x), on an interval [a,b] [ a, b]. We then rotate this curve about a given axis to get the surface of the solid of revolution. For purposes of this discussion let’s rotate the curve about the x x -axis, although it could be any vertical or horizontal axis.

How do you find the area of a solid disk?

In the case that we get a solid disk the area is, A = π(radius)2 A = π (radius) 2 where the radius will depend upon the function and the axis of rotation. In the case that we get a ring the area is,

How do you find the cross section of a solid object?

One of the easier methods for getting the cross-sectional area is to cut the object perpendicular to the axis of rotation. Doing this the cross section will be either a solid disk if the object is solid (as our above example is) or a ring if we’ve hollowed out a portion of the solid (we will see this eventually).

Example 2 Determine the volume of the solid obtained by rotating the portion of the region bounded by y = 3√x y = x 3 and y = x 4 y = x 4 that lies in the first quadrant about the y-axis. First, let’s get a graph of the bounding region and a graph of the object.

Is area a function of x x or Y Y?

Also, in both cases, whether the area is a function of x x or a function of y y will depend upon the axis of rotation as we will see. This method is often called the method of disks or the method of rings. Let’s do an example.

What is the difference between the axis of rotation and inner radius?

So, we know that the distance from the axis of rotation to the x x -axis is 4 and the distance from the x x -axis to the inner ring is x x. The inner radius must then be the difference between these two.

How do you expand a solid with cylinders?

The first cylinder will cut into the solid at x = 1 x = 1 and as we increase x x to x = 3 x = 3 we will completely cover both sides of the solid since expanding the cylinder in one direction will automatically expand it in the other direction as well. The method used in the last example is called the method of cylinders or method of shells.

What is the formula for the area of a cylinder?

The method used in the last example is called the method of cylinders or method of shells. The formula for the area in all cases will be, A = 2π(radius)(height) A = 2 π (radius) (height) There are a couple of important differences between this method and the method of rings/disks that we should note before moving on.