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

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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
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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
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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.
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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.

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

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