What is the volume of the region bounded by X?

If the region bounded by x = f (y) and x = g (y) on [ a, b ], where f (y) ≥ g (y) is revolved about the y ‐axis, then its volume ( V) is Note again that f (x) and g (x) and f (y) and g (y) represent the outer and inner radii of the washers or the distance between a point on each curve to the axis of revolution.

How do you calculate the volume of a solid bounded by Y?

The volume of the solid bounded by y = f ( x) rotating about the x -axis is V = π ∫ a b ( f ( x)) 2 d x. That is, you want to calculate π ( ∫ 0 1 x 2 d x − ∫ 0 1 x 4 d x)

How do you find the volume of a cylindrical region?

If the region bounded by x = f (y) and the y ‐axis on the interval [ a,b ], where f (y) ≥ 0, is revolved about the x ‐axis, then its volume ( V) is Note that the x and y in the integrands represent the radii of the cylindrical shells or the distance between the cylindrical shell and the axis of revolution.

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.

How to find the volume of the solid generated by revolving?

Example 1: Find the volume of the solid generated by revolving the region bounded by y = x 2 and the x ‐axis on [−2,3] about the x ‐axis. Because the x ‐axis is a boundary of the region, you can use the disk method (see Figure 1 ). Figure 1 Diagram for Example 1.

How do you find the volume of a solid shell?

Volumes by Cylindrical Shells: the Shell Method . Another method of find the volumes of solids of revolution is the shell method. It can usually find volumes that are otherwise difficult to evaluate using the Disc / Washer method. General formula: V = ∫ 2π (shell radius) (shell height) dx .