Math 201-NYB-05 Exercises
Integral Calculus
Arc Length (Length of a Curve)
Filename: E22.0) Arclength.Doc
Carl F. Gauss
Find the length of the given curve over the given interval.
1.
y  2x  3
2.
x  1,5
4.

x  13 y 2  2
4
x 
y  16
x  2,3
10.

3
2
5.
1
2 x2
8.
y  x 2  18 ln  x 
11.
2
3
 x 2  1
3
2
6.
24 xy  y 4  48
14.
1 x 5  1 x 3
y  12
5
y  13 x 3  x  14 arctan  x 
17.

9.

 
x  18 y 4  14 y 2
y  1, 4 
12.
15.
,3
 127 ,1 ; B :  109
12
y
1
2
 e x  e x 
18.
y  x2
30 xy 3  y 8  15
A:
x  0,1
x  0,1
3
x 1
y  12
x
x  0, 2
4 y 4  12 xy  3  0
A:
3
13 ; B : 2, 7
A : 1, 12
6
x  1, 2
y  2, 4
16.
3
1 x  2
y
 4
 2 
x   4,1
y
y  23 x 2  1
x  0, 4
x  0, 2 
x  2,3
13.
3
x  0,1
y  0,1
7.
3.
y  3x 2  1
271 , 2
 158 ,1 ; B :  240

y  ln  x 
x   3 , 8 
Hint: substitute u 
x2  1
Let f be a function defined on the given interval and having the given derivative f. (Such a function exists
by the Fundamental Theorem of Calculus.) Find the arclength of the graph of f.
19.
f  x 
x   2,3
x2  1
20.
f  x 
tan 2  x   1
x   23 , 34 
21
f  x 
x 1
x  25,100
Best of Luck !
Steve
ANSWERS:
1.
4
5 u
4.
4
3
7.
595
144
10.
5  18 ln
13.
17
6
16.
4
3
19.
5
2
2.
u
5.
u
 23 
u
 u
 16
u
u
85 85 8
243
 77 
3
2
u
 32 
3
2
u
54 2
8.
22
3
11.
331
120
14.
53
6
u
17.
1
2
5  14 ln
20.
1 ln
2
2
u
u
u


5 2 u
3.
10 5  2
3
6.
13
12
9.
2055
64
u
12.
e4 1
2 e2
u
15.
353
240
18.
1  12 ln  2   12 ln  3 u
21

u
u
u
4 10  2  5
5
5
5
2

u