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# Calculus II - 10

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Stewart Calculus Section 8.2

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• ### Calculus II - 10

1. 1. 8.2 Area of a Surface of Revolution y 0 a b x
2. 2. Take limit of sum of the area of thin bands(or frustums).y0 a b x
3. 3. The formula for area of frustum: = ( + ) is the slant hight , are the upper and lower radii
4. 4. The area of the surface obtained by rotatingthe curve = ( ), , about the x-axis: = ( ) + [ ( )]
5. 5. The area of the surface obtained by rotatingthe curve = ( ), , about the x-axis: = ( ) + [ ( )]Recall the arc length formula: = + [ ( )]
6. 6. Ex: Find the area of the surface obtained byrotating the arc = ,about the x-axis.
7. 7. Ex: Find the area of the surface obtained byrotating the arc = ,about the x-axis. ( )=
8. 8. Ex: Find the area of the surface obtained byrotating the arc = ,about the x-axis. ( )= = +
9. 9. Ex: Find the area of the surface obtained byrotating the arc = ,about the x-axis. ( )= = + =
10. 10. Ex: Find the area of the surface obtained byrotating the arc = from (0,0) to (1,1)about the y-axis.
11. 11. Ex: Find the area of the surface obtained byrotating the arc = from (0,0) to (1,1)about the y-axis. =
12. 12. Ex: Find the area of the surface obtained byrotating the arc = from (0,0) to (1,1)about the y-axis. = = +
13. 13. Ex: Find the area of the surface obtained byrotating the arc = from (0,0) to (1,1)about the y-axis. = = + = +
14. 14. Ex: Find the area of the surface obtained byrotating the arc = from (0,0) to (1,1)about the y-axis. = = + = + = ( )
15. 15. 8.3 Applications toPhysics and Engineering Ex: A dam has the shape of the trapezoid. The height is 20m, the width is 50m at top and 30m at bottom. Find the force on the dam due to hydrostatic pressure if the water is 4m from the top of the dam. 50m 4m 20m 30m
16. 16. = ·
17. 17. = · =
18. 18. = · = = , = / , = . /
19. 19. = · = = , = / , = . /= · . · ( )
20. 20. = · = = , = / , = . /= · . · ( )= ·
21. 21. = · = = , = / , = . /= · . · ( )= · .