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7-2 Volume: The Disk Method Objective: Find volume of a solid revolution using the disk method and washer method. Miss Battaglia AP Calculus. Rotate around the x-axis and find the volume of y= 1 / 2 x from 0 to 4. The Disk Method. - PowerPoint PPT Presentation
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7-2 Volume: The Disk & Washer Methods
Objective: Find volume of a solid revolution using the disk method and washer method.
Miss BattagliaAP Calculus
Rotate around the x-axis and find the volume of y=1/2 x from 0 to 4.
If a region in the plane is revolved about a line, the resulting solid is a solid of revolution, and the line is called the axis of revolution. The simplest such solid is a right circular cylinder or disk (formed by revolving a rectangle about an axis adjacent to one side of the rectangle.
Volume of disk = (area of disk)(width of disk)= πR2Δx
ΔV = πR2Δx
The Disk Method
The approximation of the volume of a solid becomes better as ||Δ|| 0 (n ∞).
Volume of a solid =
Volume of a Solid
Representative Element
Solid of revolution
To find the volume of a solid of revolution with the disk method, use one of the following:
Horizontal Axis of Revolution Vertical Axis of Revolution
The Disk Method
Find the volume of a solid formed by revolving the region bounded by the graph of
and the x-axis (0 < x < π) about the x-axis.
Using the Disk Method
Find the volume of a solid formed by revolving the region bounded by
and g(x)=1 about the line y=1.
Revolving About a Line that is Not a Coordinate Axis
1. y = - x + 1 revolved about the x-axis from 0 to 1
2. y = 4 – x2 revolved about the x-axis from 0 to 2
3. revolved about the x-axis from 1 to 4
4. revolved about the x-axis from 0 to 3
5. y = 2 and from -3 to 3
6. revolved about the x-axis from 0 to 4
7. , y = 0 from 0 to 3
8. y = 2x2, y=0, from 0 to 2
Try these!
The disk method can be extended to cover solids of revolution with holes by replacing the representative disk with a washer.
The Washer Method
Find the volume of the solid formed by revolving the region bounded by the graph of and about the x-axis.
Using the Washer Method
Find the volume of a solid formed by revolving the region bounded by the graph of y=x2+1, y=0, x=0, and x=1 about the y-axis.
Integrating with Respect to y, Two-Integral Case
A manufacturer drills a hole in the middle of a metal sphere of radius 5 in. The hole has a radius of 3 in. What is the volume of the resulting metal ring?
Manufacturing
Read 7.2 Page 465 #1-9 odd, 11, 13
Classwork/Homework