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)–Washer (disk with a hole) πr2h– Shell 2πrhw (w is typically dx/dy)

• Useful with y-axis rotations or x=# line rotations (everything is in terms of x)

Careful with y-axis rotations, everything must be in terms of y!!!

1. Find the area between the graph of and in [1,3]. Graphing by hand requires synthetic division to find roots.

Answers on last slide!

23 xxy 0y

.

2. Find the area bounded by the graph of and y=0.

.

322 xxy

3. Find the area between the graph of and

.

1xy3)1( xy

4. Find the area between the graph of and and y=0

.

6 xyxy

5. Find the area between the graph of and

.

23 yx 1xy

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

.

xy xy

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

.

27/2xy xy

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

.

12 xy

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

.

24 xy

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

.

xy 3 xy

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

.

xy 8x

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

.

2xy

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

.

12 xy

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