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Unformatted text preview: Section 7.5
(4x 2
1 313 27. f (x) 1) (8x 2 (4x 2 1)2 8x) 4x 2 8x 1 , so the (4x 1)2 31. Because the limit of the sum xk as the norm of the partition goes to zero will always be the length (b the interval (a, b). 32. No; the curve can be indefinitely long. Consider, for a) of length is
1/2 1 4x 2 8x 1 2 dx which evaluates, (4x 2 1)2 using NINT, to 2.1089. 0: example, the curve sin
1 3 1 + 0.5 for 0 x 28. There is a corner at x x 1. 33. (a) The fin is the hypotenuse of a right triangle with leg lengths xk and
[ 2, 2] by [ 1, 5] df dx x xk xk
1 f (xk 1) xk. Break the function into two smooth segments: y x3 x3 3x 3x 2
2 (b) lim
n k 1
n b n ( xk)2 xk 1 ( f (xk 1) xk)2 ( f (xk 1))2 5x 5x 5 5
1 2 0 2 0 1
2 x x x x 0 and 1 0 . 1 lim
n k 1
1 ( f (x))2 dx x c, where c is a a y 34. Yes. Any curve of the form y constant, has constant slope
a The length is
0 (y )2 dy
1 1, so that a 2. 1
2 (3x 2 5)2 dx 1
0 (3x 2 5)2 dx, 13.132. 1
0 (y )2 dx a 2 dx
0 which evaluates, using NINT for each part, to 29. y (1 x)2, 0 x 1 s Section 7.5 Applications from Science and Statistics
(pp. 401411) Quick Review 7.5
1 [ 0.5, 2.5] by [ 0.5, 1.5] 1. (a)
0 e x dx e x 1 1
0 1 e y x x 1 , but NINT may fail using y over the entire 0. So, split the curve x
1 (b)
1 0.632 e x dx 1.718
/2 interval because y is undefined at x into two equal segments by solving y x: x
1 . The total length is 2 4
1/4 2. (a)
0 ex 1 e
0 1 y 1
x 1 with
1 2 x (b) 3. (a) /2 dx, sin x dx
/4 cos x
/4 2 2 which evaluates, using NINT, to 30. 1.623. (b)
3 0.707 (x 2 2) dx
1 3 x 3
3 4. (a)
0 2x
0 15 (b) 15
2 5. (a)
1 x2 x
3 1 dx [0, 16] by [0, 2] y 1 3/4 x , but NINT may fail using y over the entire 4 1 ln (x 3 1) 3 1 [ln 9 ln 2] 3 1 9 ln 3 2 2 1 interval, because y is undefined at x x
2 0 0. So, use 6. (b)
7 0.501 2) sin x dx x 2)(2 x) dx y ,0 1 4 y 2: x 4y and the length is 3 2 (x
0 7 (4y 3)2 dy, which evaluates, 16.647. 7. using NINT, to (1
0 ...
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Spring '08 term at University of Florida.
 Spring '08
 GERMAN

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