How do you find the volume of a solid with curves?

For each of the following problems use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by the given curves about the given axis. Rotate the region bounded by y = √x y = x , y = 3 y = 3 and the y y -axis about the y y -axis.

How to find the volume of a solid formed by shell method?

Find the volume of the solid formed by rotating the region bounded by and about using the Shell Method. Solution Since our shells are parallel to the axis of rotation, we must consider the radius and height functions in terms of . The radius of a sample shell will be and the height of a sample shell will be .

When the region is rotated this thin slice is formed?

When the region is rotated, this thin slice forms a cylindrical shell, as pictured in part (b) of the figure. The previous section approximated a solid with lots of thin disks (or washers); we now approximate a solid with many thin cylindrical shells.

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).

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 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.