section 6.2 volumes
TRANSCRIPT
![Page 1: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/1.jpg)
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!
![Page 2: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/2.jpg)
No matter how you slice it…?
a b
f(x)
![Page 3: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/3.jpg)
a b
f(x)
No matter how you slice it…?
![Page 4: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/4.jpg)
a b
f(x)
No matter how you slice it…?
![Page 5: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/5.jpg)
a b
f(x)
No matter how you slice it…?
![Page 6: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/6.jpg)
a b
f(x)
No matter how you slice it…?
![Page 7: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/7.jpg)
a b
f(x)
No matter how you slice it…?
![Page 8: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/8.jpg)
a b
f(x)
No matter how you slice it…?
![Page 9: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/9.jpg)
a b
f(x)
No matter how you slice it…?
![Page 10: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/10.jpg)
a b
f(x)
No matter how you slice it…?
![Page 11: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/11.jpg)
a b
f(x)
No matter how you slice it…?
![Page 12: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/12.jpg)
a b
f(x)
No matter how you slice it…?
![Page 13: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/13.jpg)
a b
f(x)
No matter how you slice it…?
![Page 14: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/14.jpg)
a b
f(x)
No matter how you slice it…?
![Page 15: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/15.jpg)
a b
f(x)
No matter how you slice it…?
![Page 16: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/16.jpg)
a b
f(x)
No matter how you slice it…?
![Page 17: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/17.jpg)
a b
f(x)
No matter how you slice it…?
![Page 18: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/18.jpg)
a b
f(x)
No matter how you slice it…?
![Page 19: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/19.jpg)
a b
f(x)
No matter how you slice it…?
![Page 20: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/20.jpg)
a b
f(x)
No matter how you slice it…?
![Page 21: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/21.jpg)
a b
f(x)
No matter how you slice it…?
![Page 22: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/22.jpg)
a b
f(x)
No matter how you slice it…?
![Page 23: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/23.jpg)
a b
f(x)
No matter how you slice it…?
![Page 24: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/24.jpg)
a b
f(x)
No matter how you slice it…?
![Page 25: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/25.jpg)
a b
f(x)
No matter how you slice it…?
![Page 26: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/26.jpg)
a b
f(x)
No matter how you slice it…?
![Page 27: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/27.jpg)
a b
f(x)
No matter how you slice it…?
![Page 28: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/28.jpg)
a b
f(x)
No matter how you slice it…?
![Page 29: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/29.jpg)
a b
f(x)
No matter how you slice it…?
![Page 30: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/30.jpg)
a b
f(x)
No matter how you slice it…?
![Page 31: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/31.jpg)
a b
f(x)
No matter how you slice it…?
![Page 32: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/32.jpg)
a b
f(x)
No matter how you slice it…?
![Page 33: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/33.jpg)
a b
f(x)
No matter how you slice it…?
![Page 34: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/34.jpg)
a b
f(x)
No matter how you slice it…?
![Page 35: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/35.jpg)
a b
f(x)
No matter how you slice it…?
![Page 36: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/36.jpg)
a b
f(x)
No matter how you slice it…?
![Page 37: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/37.jpg)
No matter how you slice it…?
![Page 38: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/38.jpg)
No matter how you slice it…?
![Page 39: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/39.jpg)
No matter how you slice it…?
![Page 40: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/40.jpg)
No matter how you slice it…?
![Page 41: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/41.jpg)
No matter how you slice it…?
![Page 42: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/42.jpg)
No matter how you slice it…?
![Page 43: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/43.jpg)
No matter how you slice it…?
![Page 44: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/44.jpg)
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…?
![Page 45: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/45.jpg)
a b
f(x)
b
No matter how you slice it…?
![Page 46: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/46.jpg)
a b
f(x)
b
No matter how you slice it…?
![Page 47: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/47.jpg)
a b
f(x)
b
No matter how you slice it…?
![Page 48: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/48.jpg)
a b
f(x)
b
No matter how you slice it…?
![Page 49: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/49.jpg)
a b
f(x)
b
No matter how you slice it…?
![Page 50: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/50.jpg)
a b
f(x)
b
g(x)
b
a
b
a
b
a
dxxgxfdxxgdxxf ))()(()()(
No matter how you slice it…?
![Page 51: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/51.jpg)
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…?
![Page 52: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/52.jpg)
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
![Page 53: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/53.jpg)
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
![Page 54: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/54.jpg)
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…?
![Page 55: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/55.jpg)
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.
![Page 56: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/56.jpg)
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…?
![Page 57: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/57.jpg)
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…?
![Page 58: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/58.jpg)
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…?
![Page 59: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/59.jpg)
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…?
![Page 60: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/60.jpg)
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…?
![Page 61: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/61.jpg)
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
![Page 62: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/62.jpg)
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
![Page 63: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/63.jpg)
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
![Page 64: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/64.jpg)
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!
![Page 65: Section 6.2 Volumes](https://reader035.vdocuments.net/reader035/viewer/2022070600/577ccd541a28ab9e788c032d/html5/thumbnails/65.jpg)