128ahw7sum10

# 128ahw7sum10 - MATH 128A SUMMER 2010 HOMEWORK 7 SOLUTION...

This preview shows pages 1–2. Sign up to view the full content.

MATH 128A, SUMMER 2010, HOMEWORK 7 SOLUTION BENJAMIN JOHNSON Homework 7: Due Monday, July 19 4.4; 1d, 3d 4.5; 2a 4.6; 1d, 9 4.7; 1f, 8 section 4.4 1. Use the composite Trapezoid rule with the indicated values of n to approximate the following integrals. d. R x 0 x 2 cos xdx , n = 6 solution: Dividing the interval [0 ] into six pieces, the points are x 0 = 0, x 1 = π 6 , x 2 = π 3 , x 3 = π 2 , x 4 = 2 π 3 , x 5 = 5 π 6 , and x 6 = π . We use the formula h 2 h f ( x 0 ) + 2 n - 1 j =1 f ( x j ) + f ( x n ) i , obtaining π 12 h 0 + 2 ± π 2 36 · 3 2 + π 2 9 · 1 2 + π 2 4 · 0 + 4 π 2 9 · - 1 2 + 25 π 2 36 · - 3 2 ² - π 2 i ≈ - 6 . 42872. 3. Use the composite Simpson’s rule to approximate the integrals in Exercise 1. d. R x 0 x 2 cos xdx , n = 6 solution: Same coordinates and parameters as in the solution to Exercise 1, except we now use the formula: h 3 f ( x 0 ) + 2 n/ 2 - 1 X j =1 f ( x 2 j ) + 4 n/ 2 X j =1 f ( x 2 j - 1 ) + f ( x n ) . This gives π 18 h 0 + 2 ± π 2 18 - 4 π 2 18 ² + 4 ± 3 π 2 72 + 0 - 25 3 π 2 72 ² - π 2 i ≈ - 6 . 27487. Note: The exact value for this integral is - 2 π ≈ - 6 . 28319. section 4.5 2. Use Romberg integration to compute R 3 , 3 for the following integrals. a. R 1 - 1 (cos x ) 2 dx solution: We have R 1 , 1 = 1 - ( - 1) 2 ( cos 2 ( - 1) + cos 2 (1)) 0 . 583853. For R 2 , 1 , we get 1 2 (cos 2 ( - 1) + 2 cos 2 0 + cos 2 1) 1 . 29193 Then R 2 , 2 = 4 R 2 , 1 - R 1 , 1 3 = 4 · 1 . 29193 - 0 . 583853 3 1 . 52795. Using the recursive formula,

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

128ahw7sum10 - MATH 128A SUMMER 2010 HOMEWORK 7 SOLUTION...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online