1. FINDING VOLUME USING DISK METHOD & WASHER METHOD
CHAPTER 4
APPLICATIONS:
FINDING AREA
FINDING VOLUME USING DISK METHOD & WASHER METHOD

2. AREA
The definite integral represent the area between the function and the x-axis along the interval [a,b].

3. Area underneath f (x) along [-1,1]

4. Area underneath g(x) along [-1,1]

5. Area between f (x) and g(x) along [-1,1]
Upper curve
Lower curve

6. Example 1 Find the area bounded by the curve and the x-axis along the interval

7. Example 2 Find the area bounded by the curve and the x-axis along the interval

8. Example 3 Find the point of intersection, hence determine the area bounded by the curve and

9. AREA
The definite integral represent the area between the function and the y-axis along the interval [a,b].
Area underneath f (y) along [0,1]

10. AREA
Area underneath g(y) along [0,1]

11. AREA
Area between f(y) and g(y) along [0,1]
Right curve
Left curve

12. Example 4 Find the area of the region in graph below:

13. Example 5 Find the point of intersection, hence determine the area bounded by the curve and

14. VOLUME ; DISK METHOD
Volume of revolution are obtained by revolving a two-dimensional region about an axis of rotation.
Given
Where
Therefore, volume generated on the solid region is given by:

15. Consider the graph of a function y=f(x) in Quadrant I.
For volumes of revolution about the x-axis, the cross-sections of the solid are all circles.
Consider the graph of a function y=f(x) in Quadrant I.
The region between this graph and the x-axis, between x=1 and x=5, is revolved about the x-axis. x

16. VOLUME ; DISK METHOD For volumes of revolution about the x-axis:
Therefore, volume of revolution about the x-axis is given by:

17. Example 1 Determine the volume of the solid obtained by rotating the region bounded by and the x-axis about the x-axis. x

18. Example 2 Determine the volume of the solid obtained by rotating the region bounded by about the x-axis. x

19. Exercise 2* Find the volume of the solid of revolution formed by rotating the region bounded by the x-axis and the graph of from x=1 to x=2 about the x-axis.

20. Example 3 Determine the volume of the solid obtained by rotating the region bounded by and the x-axis about the x-axis. x

21. Consider the graph of a function x=f(y) in Quadrant I.
For volumes of revolution about the y-axis, the cross-sections of the solid are all circles.
Consider the graph of a function x=f(y) in Quadrant I.
The region between this graph and the x-axis, between y=1 and y=4, is revolved about the y-axis.

22. VOLUME ; DISK METHOD For volumes of revolution about the y-axis:
Therefore, volume of revolution about the x-axis is given by:

23. Example 3 Determine the volume of the solid obtained by rotating the region bounded by and the y-axis about the y-axis. y

24. Example 4 Determine the volume of the solid obtained by rotating the region bounded by and the y-axis about the y-axis. y

25. VOLUME ; WASHER METHOD
The washer method is essentially but two applications of the disk method.   Suppose that f(x) > g(x) > 0, for all x in the interval I =[a , b].
Further assume that f and g are bounded over that interval. We'll find the volume of the solid of revolution that results when the region bounded by the graphs of f and g over interval I  revolves about the x-axis.

26. At a point x on the x-axis, a perpendicular cross section of the solid consists of the region between two concentric circles (shaped like a washer, it's called an annulus).  
The outer circle has radius  R = f(x), and the inner circle (the hole) has radius    r = g(x). x

27. VOLUME ; WASHER METHOD For volumes of revolution about the x-axis:
Therefore, volume of revolution bounded by two lines about the x-axis is given by:

28. Example 5 Find the volume of the solid of revolution formed by rotating the finite region bounded by the about the x-axis. x

29. Example 6 Find the volume of the solid of revolution formed by rotating the finite region bounded by the about the x-axis. x

30. VOLUME ; WASHER METHOD
The washer method is essentially but two applications of the disk method.   Suppose that f(y) > g(y) > 0, for all y in the interval I =[a , b].
Further assume that f and g are bounded over that interval. We'll find the volume of the solid of revolution that results when the region bounded by the graphs of f and g over interval I  revolves about the y-axis.

31. At a point y on the y-axis, a perpendicular cross section of the solid consists of the region between two concentric circles  
The outer circle has radius  R = f(y), and the inner circle (the hole) has radius    r = g(y). x

32. VOLUME ; WASHER METHOD For volumes of revolution about the y-axis:
Therefore, volume of revolution bounded by two lines about the y-axis is given by:

33. Example 7 Find the volume of the solid of revolution formed by rotating the finite region bounded by the about the y-axis. y

34. Example 6 Find the volume of the solid of revolution formed by rotating the finite region bounded by the about the y-axis. y