chapter_06

# chapter_06 - January 27, 2005 11:45 L24-ch06 Sheet number 1...

This preview shows pages 1–4. Sign up to view the full content.

January 27, 2005 11:45 L24-ch06 Sheet number 1 Page number 230 black 230 CHAPTER 6 Integration EXERCISE SET 6.1 1. Endpoints 0 , 1 n , 2 n ,..., n 1 n , 1; using right endpoints, A n = " r 1 n + r 2 n + ··· + r n 1 n +1 # 1 n n 2 5 10 50 100 A n 0 . 853553 0 . 749739 0 . 710509 0 . 676095 0 . 671463 2. Endpoints 0 , 1 n , 2 n n 1 n , 1; using right endpoints, A n = · n n + n n +2 + n n +3 + + n 2 n 1 + 1 2 ¸ 1 n n 2 5 10 50 100 A n 0 . 583333 0 . 645635 0 . 668771 0 . 688172 0 . 690653 3. Endpoints 0 , π n , 2 π n ( n 1) π n ; using right endpoints, A n = [sin( π/n ) + sin(2 π/n )+ + sin( π ( n 1) /n ) + sin π ] π n n 2 5 10 50 100 A n 1 . 57080 1 . 93376 1 . 98352 1 . 99935 1 . 99984 4. Endpoints 0 , π 2 n , 2 π 2 n ( n 1) π 2 n , π 2 ; using right endpoints, A n = [cos( π/ 2 n ) + cos(2 2 n + cos(( n 1) 2 n ) + cos( 2)] π 2 n n 2 5 10 50 100 A n 0 . 555359 0 . 834683 0 . 919400 0 . 984204 0 . 992120 5. Endpoints 1 , n n , n n 2 n 1 n , 2; using right endpoints, A n = · n n + n n + + n 2 n 1 + 1 2 ¸ 1 n n 2 5 10 50 100 A n 0 . 583333 0 . 645635 0 . 668771 0 . 688172 0 . 690653 6. Endpoints π 2 , π 2 + π n , π 2 + 2 π n π 2 + ( n 1) π n , π 2 ; using right endpoints, A n = · cos ³ π 2 + π n ´ + cos µ π 2 + 2 π n + + cos µ π 2 + ( n 1) π n + cos ³ π 2 ´ ¸ π n n 2 5 10 50 100 A n 1 . 57080 1 . 93376 1 . 98352 1 . 99936 1 . 99985

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

View Full Document
January 27, 2005 11:45 L24-ch06 Sheet number 2 Page number 231 black Exercise Set 6.1 231 7. Endpoints 0 , 1 n , 2 n ,..., n 1 n , 1; using right endpoints, A n = s 1 µ 1 n 2 + s 1 µ 2 n 2 + ··· + s 1 µ n 1 n 2 +0 1 n n 2 5 10 50 100 A n 0 . 433013 0 . 659262 0 . 726130 0 . 774567 0 . 780106 8. Endpoints 1 , 1+ 2 n , 4 n 2( n 1) n , 1; using right endpoints, A n = s 1 µ n 2 n 2 + s 1 µ n 4 n 2 + + s 1 µ n 2 n 2 2 n n 2 5 10 50 100 A n 1 1 . 423837 1 . 518524 1 . 566097 1 . 569136 9. Endpoints 1 , 2 n , 4 n 1 2 n , 1; using right endpoints, A n = h e 1+ 2 n + e 1+ 4 n + e 1+ 6 n + ... + e 1 2 n + e 1 i 2 n n 2 5 10 50 100 A n 3 . 718281 2 . 851738 2 . 59327 2 . 39772 2 . 35040 10. Endpoints 1 , 1 n , 2 n 2 1 n , 2; using right endpoints, A n = · ln µ 1 n +ln µ 2 n + µ 2 1 n +ln2 ¸ 1 n n 2 5 10 50 100 A n 0 . 549 0 . 454 0 . 421 0 . 393 0 . 390 11. Endpoints 0 , 1 n , 2 n n 1 n , 1; using right endpoints, A n = · sin 1 µ 1 n + sin 1 µ 2 n + + sin 1 µ n 1 n + sin 1 (1) ¸ 1 n n 2 5 10 50 100 A n 1 . 04729 0 . 75089 0 . 65781 0 . 58730 0 . 57894 12. Endpoints 0 , 1 n , 2 n n 1 n , 1; using right endpoints, A n = · tan 1 µ 1 n + tan 1 µ 2 n + + tan 1 µ n 1 n + tan 1 (1) ¸ 1 n n 2 5 10 50 100 A n 0 . 62452 0 . 51569 0 . 47768 0 . 44666 0 . 44274 13. 3( x 1) 14. 5( x 2) 15. x ( x +2) 16. 3 2 ( x 1) 2
January 27, 2005 11:45 L24-ch06 Sheet number 3 Page number 232 black 232 Chapter 6 17. ( x + 3)( x 1) 18. 3 2 x ( x 2) 19. The area in Exercise 17 is always 3 less than the area in Exercise 15. The regions are identical except that the area in Exercise 15 has the extra trapezoid with vertices at (0 , 0) , (1 , 0) , (0 , 2) , (1 , 4) (with area 3). 20. (a) The region in question is a trapezoid, and the area of a trapezoid is 1 2 ( h 1 + h 2 ) w . (b) From Part (a), A 0 ( x )= 1 2 [ f ( a )+ f ( x )]+( x a ) 1 2 f 0 ( x ) = 1 2 [ f ( a f ( x x a ) 1 2 f ( x ) f ( a ) x a = f ( x ) 21. A (6) represents the area between x = 0 and x =6 ; A (3) represents the area between x =0 and x = 3; their di±erence A (6) A (3) represents the area between x = 3 and x = 6, and A (6) A (3) = 1 3 (6 3 3 3 ) = 63.

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.

## This note was uploaded on 06/10/2010 for the course MATH 200-177 taught by Professor Richardwhite during the Spring '10 term at Drexel.

### Page1 / 60

chapter_06 - January 27, 2005 11:45 L24-ch06 Sheet number 1...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online