1. Applications Of The Definite Integral
The area between two curves
The Volume of the Solid of revolution (by slicing)

2. AREA BETWEEN CURVES

3. AREA BETWEEN CURVES

4.   Determine the area of the region bounded by y = 2x2 +10 and y = 4x +16 between x = -2 and x = 5

5. Intersection: (-1,-2) and (5,4).

6. Intersection: (-1,-2) and (5,4).
Intersection: (-1,-2) and (5,4).
So, in this last example we’ve seen a case where we could use either method to find the area. 
However, the second was definitely easier.

7. Intersection points are:
y = - 1
y = 3

8. Volume of REVOLUTION
Find the Volume of revolution using the disk method
Find the volume of revolution using the washer method
Find the volume of revolution using the shell method
Find the volume of a solid with known cross sections

9. Area is only one of the applications of integration
Area is only one of the applications of integration. We can add up representative volumes in the same way we add up representative rectangles.
When we are measuring volumes of revolution, we can slice representative disks or washers.
These are a few of the many industrial uses for volumes of revolutions.

10. Disk Method

14. Washer Method

17. NOTE: Cross-section is perpendicular to the axis of rotation.
A solid obtained by revolving a region around a line.
NOTE: Cross-section is perpendicular to the axis of rotation.

18. Example:
Find the volume of the solid formed by revolving the region bounded by y = x and y = x² over the interval [0, 1] about the x – axis.

19. Example: rotate it around x = axis
Find the volume of the solid of revolution formed by rotating the finite region bounded by the graphs of about the x-axis.   


20. Volumes by Cylindrical Shells

22. Summing up the volumes of all these infinitely thin shells, we get the total volume of the solid of revolution:

24. u = x – x = 1 u=0
x = u x = 2 u=1
du = dx

25. Time to Practice !!! AGAIN

26. One More Example

29. The Volume for Solids with Known Cross Sections

30. Procedure: volume by slicing
sketch the solid and a typical cross section
find a formula for the area, A(x), of the cross section
find limits of integration
integrate A(x) to get volume

31. Find the volume of a solid whose base is the circle x2 + y2 = 4 and where cross sections perpendicular to the x-axis are all squares whose sides lie on the base of the circle.
First, find the length of a side of the square
the distance from the curve to the x-axis is half the length of the side of the square … solve for y
length of a side is :

32. Find the volume of a solid whose base is the circle x2 + y2 = 4 and where cross sections perpendicular to the x-axis are all squares whose sides lie on the base of the circle.

33. Find the volume of a solid whose base is the circle x2 + y2 = 4 and where cross sections perpendicular to the x-axis are all equilateral triangles whose sides lie on the base of the circle.

34. Find the volume of a solid whose base is the circle x2 + y2 = 4 and where cross sections perpendicular to the x-axis are all semicircles whose sides lie on the base of the circle.

35. Find the volume of a solid whose base is the circle x2 + y2 = 4 and where cross sections perpendicular to the x-axis are all Isosceles right triangles whose sides lie on the base of the circle.

40. Class work Grade
Use the following Google Doc link to submit your answers:

41. 1. CALCULATOR REQUIRED
The volume of the solid generated by revolving the first quadrant
region bounded by the curve and the lines x = ln 3 and
y = 1 about the x-axis is closest to
a) b) c) d) e) 2.91

42. 2. CALCULATOR REQUIRED

43. 3. CALCULATOR REQUIRED

44. 4. CALCULATOR REQUIRED

45. 5. NO CALCULATOR

46. 6. NO CALCULATOR