Section 6.2. Solids of Revolution – if a region in the plane is revolved about a line...
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Transcript of Section 6.2. Solids of Revolution – if a region in the plane is revolved about a line...
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
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
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 region
bounded 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
42
xy
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
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