Lecture Notes Math 150B Nguyen 1 of 6 7.2 §7.2 VOLUME: The DISK method

We’ll use the definite integral to find the volume of a 3-dimensional solid whose cross sections are similar (such as bottles, pills, axles, etc.)

Def: If a region in a plane is revolved about a line, the resulting solid is called a solid of revolution, and that line is called the axis of revolution.

The simplest such solid is a right circular cylinder, or disk, which is formed by revolving a rectangle about an axis adjacent to one side of the rectangle.

Volume of 1 representative disk = (area of disk) ( width of disk) =  R2  w

Now consider a solid of revolution by revolving the following plane region about the x-axis. Lecture Notes Math 150B Nguyen 2 of 6 7.2

This approximation appears to become better and better as   0 (meaning, as n   ). So we can define the volume of the solid as

n b Volume of solid = lim  R() x 2x   R() x 2dx  0   i1 a

A similar formula can be derived if the axis of vertical is vertical.

In summary:

Note: In the Disk method, we determine the variable of integration by placing a representative rectangle “perpendicular” to the axis of revolution. If the width of the rectangle is x, integrate with respect to x, and if the width of the rectangle is y, integrate with respect to y. Lecture Notes Math 150B Nguyen 3 of 6 7.2 Example 1: Find the volume of the solid formed by revolving about the x-axis the region under the curve y  x from x = 0 to x = 1.

Example 2: Find the volume of the solid formed by revolving the region bounded by y  2x2 , y = 0 and x = 2 about the line x = 2. Lecture Notes Math 150B Nguyen 4 of 6 7.2 The WASHER Method

The disk method can be extended to cover solids of revolution with holes by replacing the representative disk with a representative washer.

Volume of 1 representative washer = (area of washer) (width of washer) =  R2  r 2  w =  R2  r 2  w

Consider a region bounded by an outer radius R() x and an inner radius r() x . The volume of the solid of revolution is given by

b 2 2 V    R() x  r() x   dx a

Note 1: The integral involving the inner radius represents the volume of the hole and is subtracted from the integral involving the outer radius.

d 2 Note 2: If the axis of revolution is vertical, then Volume V    R() y  2 r() y   dy c Lecture Notes Math 150B Nguyen 5 of 6 7.2 Example 3: Find the volume of the solid formed by revolving the region bounded by  y  sec x , y = 0, 0 x  3 about the line y = 4. Lecture Notes Math 150B Nguyen 6 of 6 7.2 Example 4: Find the volume of the solid formed by revolving the region bounded by x2 y 2  and y = 1 4 about the x-axis.

CONCLUSION: With the Disk method, we’ve found the volume of a solid having a circular cross section whose area is A   R2 . This method can be generalized to solids of any shape, as long as we know a formula for the area of an arbitrary cross section. Some common cross sections are squares, rectangles, triangles, semicircles, and trapezoids.