Area and volume practice
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Transcript of Area and volume practice
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