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§7.2 VOLUME: the DISK Method

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§7.2 VOLUME: the DISK Method 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 2x R() x 2dx 0 i1 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..
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