The document discusses calculating the area of a surface obtained by rotating a curve around an axis. It provides the general formulas for finding the area which involves integrating the function squared plus the integral of its derivative. Several examples are worked out for rotating curves and arcs around the x-axis and y-axis. The document also gives an example problem of finding the force on a dam due to hydrostatic water pressure.

1. 8.2 Area of a Surface of
Revolution
y
0 a b x

2. Take limit of sum of the area of thin bands
(or frustums).
y
0 a b x

3. The formula for area of frustum:
= ( + )
is the slant hight
, are the upper and lower radii

4. The area of the surface obtained by rotating
the curve = ( ), , about the x-axis:
= ( ) + [ ( )]

5. The area of the surface obtained by rotating
the curve = ( ), , about the x-axis:
= ( ) + [ ( )]
Recall the arc length formula:
= + [ ( )]

6. Ex: Find the area of the surface obtained by
rotating the arc = ,
about the x-axis.

7. Ex: Find the area of the surface obtained by
rotating the arc = ,
about the x-axis.
( )=

8. Ex: Find the area of the surface obtained by
rotating the arc = ,
about the x-axis.
( )=
= +

9. Ex: Find the area of the surface obtained by
rotating the arc = ,
about the x-axis.
( )=
= +
=

10. Ex: Find the area of the surface obtained by
rotating the arc = from (0,0) to (1,1)
about the y-axis.

11. Ex: Find the area of the surface obtained by
rotating the arc = from (0,0) to (1,1)
about the y-axis.
=

12. Ex: Find the area of the surface obtained by
rotating the arc = from (0,0) to (1,1)
about the y-axis.
=
= +

13. Ex: Find the area of the surface obtained by
rotating the arc = from (0,0) to (1,1)
about the y-axis.
=
= +
= +

14. Ex: Find the area of the surface obtained by
rotating the arc = from (0,0) to (1,1)
about the y-axis.
=
= +
= +
= ( )

15. 8.3 Applications to
Physics 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. = ·

17. = ·
=

18. = ·
=
= , = / , = . /

19. = ·
=
= , = / , = . /
= · . · ( )

20. = ·
=
= , = / , = . /
= · . · ( )
= ·

21. = ·
=
= , = / , = . /
= · . · ( )
= ·
.