SECTION 8.1
√
2. Use the arc length formula with y = 2 − x2 ,
dy
−x
=√
.
dx
2 − x2
Hence
Z 1r
dy
1 + ( )2 dx
L =
dx
0
Z 1r
x2
1+
dx
=
2 − x2
0
Z 1 √
2
√
=
dx
2 − x2
0
¯1
√
¯
−1 x
=
2[sin ( √ )]¯
2 0
√
2
π.
=
4
√
The curve is one-eight of the circle with radius 2, and the length is
√
1 √
2
.2. 2π =
π.
8
4
10. Differentiating the equation implicitly with respect to y,
dx
1
= (y 3 − y −3 ).
dy
2
Hence
Z 2s
dx
L =
1 + ( )2 dy
dy
1
r
Z 2
1
1 + (y 6 − 2 + y −6 )dy
=
4
1
Z 2r
1 6
=
(y + 2 + y −6 )dy
4
1
Z
1 2 3
=
(y + y −3 )dy
2 1
1 y 4 y −2 ¯¯2
[ +
]¯
=
2 4
−2 1
33
=
.
16
The length of the curve is 33
.
16
2
SECTION 8.1
SECTION 8.1
3