Unformatted text preview: 15. The volume remaining is
b V = 2
a xh 1  x2 x3  2 3b x b
b a dx = 2 h a3 2 = h(b2  a 2 )  h b2  3 b 3 1 2a = h b2  3a 2 + cu. units. 3 b
y h x y b + h =1 x=a x dx b x Fig. 115 6. Rotate about a) the xaxis
1 V =
0 (x 2  x 4 ) d x
1 1 1 = x3  x5 3 5 b) the yaxis
1 =
0 2 cu. units. 15 V = 2
0 x(x  x 2 ) d x
1 1 1 = 2 x 3  x 4 3 4
y =
0 cu. units. 6 (1,1) y=x y=x 2 x Fig. 16 4. Slicing:
1 V =
0 (y  y 4 ) dy
1 1 1 = y2  y5 2 5 Shells: V = 2
0 1 =
0 3 cu. units. 10 x(x 1/2  x 2 ) d x
1 2 1 = 2 x 5/2  x 4 5 4
y =
0 3 cu. units. 10 (1,1) y= x y=x 2 x Fig. 14 1. By slicing:
1 V =
0 x4 dx = cu. units. 5 By shells:
1 V = 2
0 y(1  y) dy
1 = 2 y2 2y 5/2  2 5
y y=x 2 =
0 cu. units. 5 (1,1) x x Fig. 11 ...
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 Fall '06
 GROSS
 Calculus, Trigraph, cu. units

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