2. Section ∆ Volumes
Solids of Revolution
Two types:
Disks
Washers

3. Disks – the theory
revolving function around an axis creating a SOLID piece
formal definition: let S be a solid that lies between x = a and x = b. If the cross-sectional area S in the plane Px, through x and the perpendicular to the x-axis, is A(x), where A is a continuous function, then the volume of S is:
don’t forget:
same rules apply,
when switching x and y!

4. Disks – the application
How do we use this?
we use circular cross-sections
A=πr2
Use “c” as the equation for the line of revolution; on the axes, c=0
So…
Horizontally
Vertical axis

5. Example 1: around the x-axis (horizontal)
Volume

6. Example 2: horizontal line of revolution
rotated about y = 1
Volume

7. Example 3: around the y-axis (vertical)
rotated about y-axis
converting x to y:
Volume

8. Washers similar idea, but with a twist: a hole
essentially, you do the same thing twice
Outer function
Inner function

9. Example 4: horizontal washer
rotated about y = 2
Volume

10. Example 5: vertical washer
rotated about x = 2
convert:
Volume

11. Example 6: complex integration
rotated about y = 3
Volume
sometimes:
you just KNOW it’s going to be ugly…
so:
cheat! use your calculator!!

12. Homework: Pg. 423 (1-11)all *(1-6) Set up only.
*(7-10) Set up and integrate by hand.
*(11) graph on graph paper 4 times and include new lines of revolution. Set up and integrate by hand..