x2 t05 02 cylindrical shells (2012)
TRANSCRIPT
![Page 1: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/1.jpg)
rotation||slice
Volumes By Cylindrical Shells
![Page 2: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/2.jpg)
rotation||slice
Volumes By Cylindrical Shellsy
x
xfy
![Page 3: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/3.jpg)
rotation||slice
Volumes By Cylindrical Shellsy
x
xfy
a b
![Page 4: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/4.jpg)
rotation||slice
Volumes By Cylindrical Shellsy
x
xfy
a x b
![Page 5: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/5.jpg)
rotation||slice
Volumes By Cylindrical Shellsy
x
xfy
a x b
![Page 6: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/6.jpg)
rotation||slice
Volumes By Cylindrical Shellsy
x
xfy
a x b
![Page 7: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/7.jpg)
rotation||slice
Volumes By Cylindrical Shellsy
x
xfy
a xx
b
![Page 8: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/8.jpg)
rotation||slice
Volumes By Cylindrical Shellsy
x
xfy
a x
y
xb
![Page 9: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/9.jpg)
rotation||slice
Volumes By Cylindrical Shellsy
x
xfy
a x
y
x
xf
b
![Page 10: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/10.jpg)
rotation||slice
Volumes By Cylindrical Shellsy
x
xfy
a x
y
x
xf
x2b
![Page 11: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/11.jpg)
rotation||slice
Volumes By Cylindrical Shellsy
x
xfy
a x
y
x
xf
xfxxA 2
x2b
![Page 12: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/12.jpg)
rotation||slice
Volumes By Cylindrical Shellsy
x
xfy
a x
y
x
xf
xfxxA 2 xxfxV 2
x2b
![Page 13: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/13.jpg)
rotation||slice
Volumes By Cylindrical Shellsy
x
xfy
a x
y
x
xf
xfxxA 2 xxfxV 2
b
axxxxfxV 2lim
0
x2b
![Page 14: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/14.jpg)
rotation||slice
Volumes By Cylindrical Shellsy
x
xfy
a x
y
x
xf
xfxxA 2 xxfxV 2
b
axxxxfxV 2lim
0
b
a
dxxxf2
x2b
![Page 15: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/15.jpg)
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
![Page 16: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/16.jpg)
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2
y4
![Page 17: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/17.jpg)
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2
y
y4
![Page 18: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/18.jpg)
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2
y
y4
![Page 19: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/19.jpg)
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2
y
y4
y
![Page 20: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/20.jpg)
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2
y
y4
y
x2
21
4 yx
![Page 21: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/21.jpg)
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2
y
y4
y
x2
21
4 yx
![Page 22: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/22.jpg)
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2
y
y4
y
x2
21
4 yx
21
42 y
![Page 23: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/23.jpg)
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2
y
y4
y
x2
21
4 yx
21
42 y
y2
![Page 24: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/24.jpg)
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2
y
y4
y
x2
2
1422 yyyA
21
4 yx
21
42 y
y2
![Page 25: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/25.jpg)
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2
y
y4
y
x2
2
1422 yyyA
yyyV 21
44
21
4 yx
21
42 y
y2
![Page 26: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/26.jpg)
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2
y
y4
y
x2
2
1422 yyyA
yyyV 21
44
4
0
21
044lim
xyyyyV
21
4 yx
21
42 y
y2
![Page 27: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/27.jpg)
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2
y
y4
y
x2
2
1422 yyyA
yyyV 21
44
4
0
21
044lim
xyyyyV
4
0
21
44 dyyy 21
4 yx
21
42 y
y2
![Page 28: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/28.jpg)
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2
y
y4
y
x2
2
1422 yyyA
yyyV 21
44
4
0
21
044lim
xyyyyV
4
0
21
44 dyyy 21
4 yx
21
42 y
y2
4
0
21
44 dyyy
![Page 29: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/29.jpg)
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2
y
y4
y
x2
2
1422 yyyA
yyyV 21
44
4
0
21
044lim
xyyyyV
4
0
21
44 dyyy 21
4 yx
21
42 y
y2
4
0
21
44 dyyy
4
0
23
21
44 dyyy
![Page 30: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/30.jpg)
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2
y
y4
y
x2
2
1422 yyyA
yyyV 21
44
4
0
21
044lim
xyyyyV
4
0
21
44 dyyy
4
0
25
23
52
384
yy
21
4 yx
21
42 y
y2
4
0
21
44 dyyy
4
0
23
21
44 dyyy
![Page 31: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/31.jpg)
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2
y
y4
y
x2
2
1422 yyyA
yyyV 21
44
4
0
21
044lim
xyyyyV
4
0
21
44 dyyy
4
0
25
23
52
384
yy
3units 15
512
032528
384
21
4 yx
21
42 y
y2
4
0
21
44 dyyy
4
0
23
21
44 dyyy
![Page 32: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/32.jpg)
4line about the
rotated is 4 when generated volume theFind 22
xyxii
![Page 33: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/33.jpg)
4line about the
rotated is 4 when generated volume theFind 22
xyxii
x
422 yx
y
2
-2
2-2
![Page 34: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/34.jpg)
4line about the
rotated is 4 when generated volume theFind 22
xyxii
x
422 yx
4
y
2
-2
2-2
![Page 35: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/35.jpg)
4line about the
rotated is 4 when generated volume theFind 22
xyxii
x
422 yx
4
x
y
2
-2
2-2
![Page 36: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/36.jpg)
4line about the
rotated is 4 when generated volume theFind 22
xyxii
x
422 yx
4
x
y
2
-2
2-2
![Page 37: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/37.jpg)
4line about the
rotated is 4 when generated volume theFind 22
xyxii
x
422 yx
4
x
y
2
-2
2-2
![Page 38: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/38.jpg)
4line about the
rotated is 4 when generated volume theFind 22
xyxii
x
422 yx
4
x
y
2
-2
2-2
torus
![Page 39: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/39.jpg)
4line about the
rotated is 4 when generated volume theFind 22
xyxii
x
422 yx
4
x
y
2
-2
2-2
torus
![Page 40: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/40.jpg)
4line about the
rotated is 4 when generated volume theFind 22
xyxii
x
422 yx
4
x
y
2
-2
2-2
torus
x
![Page 41: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/41.jpg)
4line about the
rotated is 4 when generated volume theFind 22
xyxii
x
422 yx
4
x
y
2
-2
2-2
torus
x
y2
![Page 42: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/42.jpg)
4line about the
rotated is 4 when generated volume theFind 22
xyxii
x
422 yx
4
x
y
2
-2
2-2
torus
x
y2242 x
![Page 43: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/43.jpg)
4line about the
rotated is 4 when generated volume theFind 22
xyxii
x
422 yx
4
x
y
2
-2
2-2
torus
x
y2
x42
242 x
![Page 44: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/44.jpg)
4line about the
rotated is 4 when generated volume theFind 22
xyxii
x
422 yx
4
x
y
2
-2
2-2
torus
x
y2
x42
242 x
xxxA 4242 2
![Page 45: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/45.jpg)
4line about the
rotated is 4 when generated volume theFind 22
xyxii
x
422 yx
4
x
y
2
-2
2-2
torus
x
y2
x42
242 x
xxxA 4242 2
2444 xx
![Page 46: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/46.jpg)
4line about the
rotated is 4 when generated volume theFind 22
xyxii
x
422 yx
4
x
y
2
-2
2-2
torus
x
y2
x42
242 x
xxxA 4242 2
xxxV 2444
2444 xx
![Page 47: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/47.jpg)
2
2
2
0444lim
xxxxxV
![Page 48: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/48.jpg)
2
2
2
0444lim
xxxxxV
2
2
2444 dxxx
![Page 49: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/49.jpg)
2
2
2
0444lim
xxxxxV
2
2
2444 dxxx
2
2
22
2
2 44444 dxxxdxx
![Page 50: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/50.jpg)
2
2
2
0444lim
xxxxxV
2
2
2444 dxxx
2
2
22
2
2 44444 dxxxdxx
2
2
22
2
2 44416 dxxxdxx
![Page 51: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/51.jpg)
2
2
2
0444lim
xxxxxV
2
2
2444 dxxx
2
2
22
2
2 44444 dxxxdxx
2
2
22
2
2 44416 dxxxdxx
semi-circle
![Page 52: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/52.jpg)
2
2
2
0444lim
xxxxxV
2
2
2444 dxxx
222116
2
2
22
2
2 44444 dxxxdxx
2
2
22
2
2 44416 dxxxdxx
semi-circle
![Page 53: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/53.jpg)
2
2
2
0444lim
xxxxxV
2
2
2444 dxxx
222116
2
2
22
2
2 44444 dxxxdxx
2
2
22
2
2 44416 dxxxdxx
semi-circle function oddeven odd
![Page 54: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/54.jpg)
2
2
2
0444lim
xxxxxV
2
2
2444 dxxx
222116
2
2
22
2
2 44444 dxxxdxx
2
2
22
2
2 44416 dxxxdxx
semi-circle function oddeven odd
0
![Page 55: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/55.jpg)
2
2
2
0444lim
xxxxxV
2
2
2444 dxxx
222116
2
2
22
2
2 44444 dxxxdxx
2
2
22
2
2 44416 dxxxdxx
semi-circle function oddeven odd
0
32units32
![Page 56: X2 t05 02 cylindrical shells (2012)](https://reader034.vdocuments.site/reader034/viewer/2022052523/556f5e67d8b42a916b8b510a/html5/thumbnails/56.jpg)
2
2
2
0444lim
xxxxxV
2
2
2444 dxxx
222116
Exercise 3B; 3, 4, 6, 8
Exercise 3C;1, 3, 9, 12, 20
2
2
22
2
2 44444 dxxxdxx
2
2
22
2
2 44416 dxxxdxx
semi-circle function oddeven odd
0
32units32