Slide 5.3.pptx - Solid of Revolution Disk Method Disk Method Let\u2019s rotate the region under f(x from a to b about x-axis Separate the area into

Slide 5.3.pptx - Solid of Revolution Disk Method Disk...

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Solid of Revolution. Disk Method
Disk Method Let’s rotate the region under f(x) from a to b about x-axis. Separate the area into rectangles of equal length. Revolve each separately to form many disk. V of 1 disk = π(radius) 2 •(thickness) Total volume = sum of volumes of infinitely thin individual disks.
Formulas
Example 1 Find the volume of the solid generated when the region bounded by y = 3x - x 2 and the x-axis is revolved around the x-axis. 1. Graph the region and identify the thickness (dx or dy) 2. Express the radius for one disk in terms of x (if dx) or y (if dy). 3. Integrate: Use Disk Method Since the region is revolved around x- axis → Thickness of a disk = dx. 81π/10
Example 2The region bounded by the y-axis and the graphs of y = x3/8, y = 0, and y = 8 is revolved about the y-axis. Find the volume of the resulting solid.Thickness of disks: dyRadius of disks:Why Because ?
Example 3 Sketch the region R bounded by the graph of y = sin(2x) and y = 0 from 0 to π, and find the volume of the solid generated if R is revolved around the x-axis. As the graph of y = sin(2x) is symmetric with respect to the point (pi/2,0), the area of R is 2* Integral (x from 0 to pi/2) sin 2 (x) = (1-cos(2x))/2

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