amp Washer Methods Objective Find volume of a solid revolution using the disk method and washer method Miss Battaglia AP Calculus Rotate around the xaxis and find the volume of y 1 2 ID: 390538
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Slide1
7-2 Volume: The Disk & Washer MethodsObjective: Find volume of a solid revolution using the disk method and washer method.
Miss
Battaglia
AP CalculusSlide2
Rotate around the x-axis and find the volume of y=1/2 x from 0 to 4.Slide3
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)= πR2ΔxΔV = πR2Δx
The Disk MethodSlide4Slide5
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 revolutionSlide6
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 MethodSlide7
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 MethodSlide8
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 AxisSlide9
1. y = - x + 1 revolved about the x-axis from 0 to 12. y = 4 – x
2
revolved about the x-axis from 0 to 2
3. revolved about the x-axis from 1 to 44. revolved about the x-axis from 0 to 35. y = 2 and from -3 to 36. revolved about the x-axis from 0 to 47. , y = 0 from 0 to 3
8. y = 2x
2
, y=0, from 0 to 2
Try these!Slide10
The disk method can be extended to cover solids of revolution with holes by replacing the representative disk with a washer.
The Washer MethodSlide11
Find the volume of the solid formed by revolving the region bounded by the graph of and about the x-axis.
Using the Washer MethodSlide12
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 CaseSlide13
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?
ManufacturingSlide14
Read 7.2 Page 465 #1-9 odd, 11, 13Classwork/Homework