1. Section 6.2, Part 1 Volumes: D isk Method AP Calculus December 4, 2009 Berkley High School, D2B2 todd1@toddfadoir.com

2. Calculus, Section 6.2, Todd Fadoir, CASA, 20032 First, Areas The integral represents…  The summation The x 2 represents…  the height of the rectangle the dx represents…  the width of the rectangle The product of x 2 and dx represents…  the area of a rectangle

3. Calculus, Section 6.2, Todd Fadoir, CASA, 20033 Next, Volumes What if we twirled an area about a axis? For example, happens when we take the area bounded by y=x 2, y=0, and x=1 and rotate it around the x- axis?

4. Calculus, Section 6.2, Todd Fadoir, CASA, 20034 Next, Volumes What if we twirled an area about an axis? For example, happens when we take the area bounded by y=x 2, y=0, and x=1 and rotate it around the x- axis?

5. Calculus, Section 6.2, Todd Fadoir, CASA, 20035 Next, Volumes What if we twirled an area about a axis? For example, happens when we take the area bounded by y=x 2, y=0, and x=1 and rotate it around the x- axis?

6. Calculus, Section 6.2, Todd Fadoir, CASA, 20036 Volumes We might envision each rectangle from the area example being twirled around the x-axis. What shape is formed from rotating a sample rectangle?  A disk or a short, squat cylinder

7. Calculus, Section 6.2, Todd Fadoir, CASA, 20037 Volumes What are the dimensions of the cylinder formed?  radius = the distance from the x-axis to the curve  height = dx

8. Calculus, Section 6.2, Todd Fadoir, CASA, 20038 Volumes What is the formula for the volume of a disk?  V=πr 2 h  V=πf(x) 2 dx Then could we add together (sum) all the little volume…

9. Calculus, Section 6.2, Todd Fadoir, CASA, 20039 Assignment Section 6.2, 1, 3 But wait, there’s more…

10. Calculus, Section 6.2, Todd Fadoir, CASA, 200310 Imagine an area bounded by y=x^.5, y=1, and the y-axis.

11. Calculus, Section 6.2, Todd Fadoir, CASA, 200311 Imagine an area bounded by y=x^.5, y=1, and the y-axis rotated around the y-axis.

12. Calculus, Section 6.2, Todd Fadoir, CASA, 200312 Are the disks “cutting” through the x-axis or the y-axis?  y-axis

13. Calculus, Section 6.2, Todd Fadoir, CASA, 200313 How does this effect the choice of “dx” or “dy”?  We choose “dy” because the disk cuts the y-axis.

14. Calculus, Section 6.2, Todd Fadoir, CASA, 200314 Trick: Since the integral uses “dy”, the expression which describes the area must be a function of y

15. Calculus, Section 6.2, Todd Fadoir, CASA, 200315 Think of some arbitrary y on the interval [0,1]. What is an expression for the radius of the disk?

16. Calculus, Section 6.2, Todd Fadoir, CASA, 200316

17. Calculus, Section 6.2, Todd Fadoir, CASA, 200317 (y 2,y)

18. Calculus, Section 6.2, Todd Fadoir, CASA, 200318 What is the volume of a solid generated by rotating the area between the curves y=x^.5 and y=x 2 about the y-axis?

19. Calculus, Section 6.2, Todd Fadoir, CASA, 200319 dx or dy?  dy because the disk cuts through the y-axis

20. Calculus, Section 6.2, Todd Fadoir, CASA, 200320 Our cross sectional area now is a circle with a hole in it

21. Calculus, Section 6.2, Todd Fadoir, CASA, 200321 Let’s choose an arbitrary y on the interval [0,1]

22. Calculus, Section 6.2, Todd Fadoir, CASA, 200322 What’s an expression to describe the cross- sectional area?

23. Calculus, Section 6.2, Todd Fadoir, CASA, 200323

24. Calculus, Section 6.2, Todd Fadoir, CASA, 200324 Assignment Section 6.2, 5-9, all