1. Review-Unit 6
• Area
– With respect to the x-axis
– With respect to the y-axis (Careful with y-axis rotations,
everything must be in terms of y!!!)
– Inverse (gets everything back in terms of x)
• Useful when you see x=
• Also helpful with square root functions sometimes
• Volume
– Disk πr2h (h is typically dx/dy) Careful with y-axis
rotations, everything
– Washer (disk with a hole) πr2h must be in terms of
y!!!
– Shell 2πrhw (w is typically dx/dy)
• Useful with y-axis rotations or x=# line rotations
(everything is in terms of x)

2. .
1. Find the area between the
graph of y = x − x + 2 and y = 0
3
in [1,3]. Graphing by hand requires
synthetic division to find roots.
Answers on last slide!

3. .
2. Find the area bounded by the
graph of y = x + 2 x − 3 and y=0.
2

4. .
3. Find the area between the
graph of y = ( x − 1) and y = x − 1
3

5. .
4. Find the area between the
graph of and y = − x + 6 and y=0
y= x

6. .
5. Find the area between the
graph of y = x − 1 and x = 3 − y 2

7. .
6. Find the volume of the region
bounded by the curves y = x and
y = x and rotated about the y-axis

8. .
7. Find the volume of the region
bounded by the curves y = x / 27 and
2
y = x and rotated about the x-axis

9. .
8. Find the volume of the region
bounded by the curves y = x + 1 , x=0
2
and y=0 and x=2 rotated about the
x-axis

10. .
9. Find the volume of the region
bounded by the curves y = 4 − x , x=0
2
and y=0 (Q I) rotated about the x-axis

11. .
10. Find the volume of the region
bounded by the curves y = x and y = x
3
(Q I) rotated about the y-axis

12. .
11. Find the volume of the region
bounded by the curves y = x and x = 8
(Q I) rotated about the line x=8

13. .
12. Find the volume of the region
bounded by the curves y = x and
2
y=2x (Q I) rotated about the y-axis

14. .
13. Find the volume of the region
bounded by the curves y = x + 1 ,
2
x=0, x=5 and y=0 (Q I) rotated about
the y-axis

15. Answers:
1) 20 u2 8) 206∏/15 u3
2) 32/3 u2 9) 53.62 u3
3) ½ u2 10) .18 u3
4) 7.33 u2 11) 303.3 u3
5) 9/2 u2 12) 8∏/3 u3
6) 2∏/15 ≈ .419 u3 13) 675∏/2 u3
7) 243∏/10 u3