Transcript
No matter how you slice it…?
Objectives (what you gotta do)
1.Given a function and axis of rotation…a)Draw a solid of
revolutionb)Determine the volume!
No matter how you slice it…?
a b
f(x)
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
a b
f(x)
No matter how you slice it…?
No matter how you slice it…?
No matter how you slice it…?
No matter how you slice it…?
No matter how you slice it…?
No matter how you slice it…?
No matter how you slice it…?
No matter how you slice it…?
hrVolume 2
xxf
xr
hrVolume
2
2
2
)(
xr
hrVolume
2
2
b
a
n
kknxxfdxxf
1
)(lim)(
b
a
n
kkn
xxfdxxf1
22 )(lim)(
r
No matter how you slice it…?
a b
f(x)
b
No matter how you slice it…?
a b
f(x)
b
No matter how you slice it…?
a b
f(x)
b
No matter how you slice it…?
a b
f(x)
b
No matter how you slice it…?
a b
f(x)
b
No matter how you slice it…?
a b
f(x)
b
g(x)
b
a
b
a
b
a
dxxgxfdxxgdxxf ))()(()()(
No matter how you slice it…?
a b
f(x)
b
g(x)
b
a
b
a
b
a
dxxgxfdxxgdxxf ))()(()()( 2222
b
a
b
a
b
a
dxxgxfdxxgdxxf ))()(()()(
No matter how you slice it…?
No matter how you slice it…?
Find the volume of the solid obtained by rotating the curve from x=0 to x=1.
xy
1
0
2)( dxx
2
No matter how you slice it…?
Find the volume of the solid obtained by rotating the curve from x=0 to x=ln2.
xexf )(
2ln
0
2)( dxex
23
Find the volume of the solid obtained by rotating the region bounded by y=x3, y=8 and y=0 about the y-axis.
No matter how you slice it…?
No matter how you slice it…?
Find the volume of the solid obtained by rotating the region bounded by y=x3, y=8 and y=0 about the y-axis.
Find the volume of the solid obtained by rotating the region bounded by y=x3, y=8 about the y-axis.
No matter how you slice it…?
Find the volume of the solid obtained by rotating the region bounded by y=x3, y=8 about the y-axis.
No matter how you slice it…?
Find the volume of the solid obtained by rotating the region bounded by y=x3, y=8 about the y-axis.
No matter how you slice it…?
Find the volume of the solid obtained by rotating the region bounded by y=x3, y=8 about the y-axis.
No matter how you slice it…?
Find the volume of the solid obtained by rotating the region bounded by y=x3, y=8 about the y-axis.
8
0
2)( 31
dyy
596
No matter how you slice it…?
Find the volume of the solid obtained by rotating the region between the curves y=0 and y=x2 around the line y=-3 from x=0 to x=4.
No matter how you slice it…?
y=-3
Find the volume of the solid generated by revolving the area bounded by y=-2, y=5, x=2 and y=2x-6 around the axis x=-3.
y
x
Find the volume of the solid obtained by rotating the region between the curves y=x and y=x2 around the line y=2.
1
0
222 )2()2( dxxx
158
No matter how you slice it…?
y=2
It matters how you slice it…!!It matters how you slice it…!!1.Figure out how to slice
it!2.Determine the
area/volume/length of that one piece based on the functions involved!
3.Generate a Riemann Sum
4.Let the number of pieces approach infinity!
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