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Volume – Disc & Washer Methods & Cross Sections. Section 6.2. Volume – Disc Method. Solids of Revolution – if a region in the plane is revolved about a line “line-axis of revolution” Simplest Solid – right circular cylinder or “Disc” Volume : circular cylinder = π r 2 h. - PowerPoint PPT Presentation
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Volume – Disc & Washer Methods & Cross Sections
Section 6.2
Volume – Disc Method Solids of Revolution –
if a region in the plane is revolved about a line “line-axis of revolution”
Simplest Solid – right circular cylinder or “Disc”
Volume: circular cylinder = πr2h
Disc Method – representative rectangle is perpendicular to axis (and touches axis of rotation)
i) Horizontal Axis of Revolution
2
Rotate about the x-axis
b
aV R x dx
2
Rotate about the -axis
d
cV R y dy
y
i) Vertical Axis of Revolution
2 2volume cylinder r h r w
2 2
1
limn b
i an i
R x x R x dx
Homework P.430 # 1-5,15
Washer Method Representative rectangle is perpendicular to the
axis of revolution (does NOT touch the axis) Solid of Revolution with a hole
2 2d
cV R r dy 2 2b
aV R r dx
Washer Method Outer radius – inner radius
2 2
2 2
b b
a a
b
a
V R x dx r x dx
R x r x dx
Practice Problem 1 Find the volume of the solid generated by revolving the region
bounded by the graph of y=x3, y=1, and x=2 about the x-axis.
Practice Problem 2Find the volume of the solid generated by revolving the regionbounded by the graph of y=x3, y=x, and between x=0 and x=1, about the y-axis.
Practice Problem 3 Find the volume of the solid formed by revolving the region
bounded by the graphs y=4x2 and y=16 about the line y=16.
Practice Problem 4 Find the volume of the solid formed by revolving the region
bounded by the graphs y=2 and about the line y=1.2
42xy
Practice Problem 5 Find the volume of the solid formed by revolving the region
bounded by the graphs y=0, x=1 and x=4 about the line y=4,y x
Homework P.430 # 11, 16, 17, 19, 23, 27, 32, 34
Cross Sections
1.
to x-axis
b
aA x dx
2.
to y-axis
d
cA y dy
Represents the Area of the cross sectionA x
Area of an Equilateral Triangle
2 34sA
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