Calculus: Early Transcendentals, by Anton, 7th Edition,ch06

Calculus - Early Transcendentals

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223 CHAPTER 6 Integration EXERCISE SET 6.1 1. Endpoints 0 , 1 n , 2 n , . . . , n 1 n , 1; using right endpoints, A n = 1 n + 2 n + · · · + 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 + 1 + 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 , π ; 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 π/ 2 n ) + · · · + cos(( n 1) π/ 2 n ) + cos( π/ 2)] π 2 n n 2 5 10 50 100 A n 0 . 555359 0 . 834683 0 . 919405 0 . 984204 0 . 992120 5. Endpoints 1 , n + 1 n , n + 2 n , . . . , 2 n 1 n , 2; using right endpoints, A n = n n + 1 + n n + 2 + · · · + 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 π 2 + π n + cos π 2 + 2 π n + · · · + cos π 2 + ( n 1) π n + cos π 2 π n n 2 5 10 50 100 A n 1 . 99985 1 . 93376 1 . 98352 1 . 99936 1 . 99985
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224 Chapter 6 7. Endpoints 0 , 1 n , 2 n , . . . , n 1 n , 1; using right endpoints, A n = 1 1 n 2 + 1 2 n 2 + · · · + 1 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 , 1 + 4 n , . . . , 1 + 2( n 1) n , 1; using right endpoints, A n = 1 n 2 n 2 + 1 n 4 n 2 + · · · + 1 n 2 n 2 + 0 2 n n 2 5 10 50 100 A n 1 1 . 423837 1 . 518524 1 . 566097 1 . 569136 9. 3( x 1) 10. 5( x 2) 11. x ( x + 2) 12. 3 2 ( x 1) 2 13. ( x + 3)( x 1) 14. 3 2 x ( x 2) 15. The area in Exercise 13 is always 3 less than the area in Exercise 11. The regions are identical except that the area in Exercise 11 has the extra trapezoid with vertices at (0 , 0) , (1 , 0) , (0 , 2) , (1 , 4) (with area 3). 16. (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 ( x ) = 1 2 [ f ( a ) + f ( x )] + ( x a ) 1 2 f ( x ) = 1 2 [ f ( a ) + f ( x )] + ( x a ) 1 2 f ( x ) f ( a ) x a = f ( x ) 17. B is also the area between the graph of f ( x ) = x and the interval [0 , 1] on the y axis, so A + B is the area of the square. 18. If the plane is rotated about the line y = x then A becomes B and vice versa. EXERCISE SET 6.2 1. (a) x 1 + x 2 dx = 1 + x 2 + C (b) ( x + 1) e x dx = xe x + C 2. (a) d dx (sin x x cos x + C ) = cos x cos x + x sin x = x sin x (b) d dx x 1 x 2 + C = 1 x 2 + x 2 / 1 x 2 1 x 2 = 1 (1 x 2 ) 3 / 2 3. d dx x 3 + 5 = 3 x 2 2 x 3 + 5 so 3 x 2 2 x 3 + 5 dx = x 3 + 5 + C
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Exercise Set 6.2 225 4. d dx x x 2 + 3 = 3 x 2 ( x 2 + 3) 2 so 3 x 2 ( x 2 + 3) 2 dx = x x 2 + 3 + C 5. d dx sin ( 2 x ) = cos (2 x ) x so cos (2 x ) x dx = sin ( 2 x ) + C 6. d dx [sin x x cos x ] = x sin x so x sin x dx = sin x x cos x + C 7. (a) x 9 / 9 + C (b) 7 12 x 12 / 7 + C (c) 2 9 x 9 / 2 + C 8. (a) 3 5 x 5 / 3 + C (b) 1 5 x 5 + C = 1 5 x 5 + C (c) 8 x 1 / 8 + C 9. (a) 1 2 x 3 dx = 1 4 x 2 + C (b) u 4 / 4 u 2 + 7 u + C 10. 3 5 x 5 / 3 5 x 4 / 5 + 4 x + C 11. ( x 3 + x 1 / 2 3 x 1 / 4 + x 2 ) dx = 1 2 x 2 + 2 3 x 3 / 2 12 5 x 5 / 4 + 1 3 x 3 + C 12. (7 y 3 / 4 y 1 / 3 + 4 y 1 / 2 ) dy = 28 y 1 / 4 3 4 y 4 / 3 + 8 3 y 3 / 2 + C 13. ( x + x 4 ) dx = x 2 / 2 + x 5 / 5 + C 14. (4 + 4 y 2 + y 4 ) dy = 4 y + 4 3 y 3 + 1 5 y 5 + C 15. x 1 / 3 (4 4 x + x 2 ) dx = (4 x 1 / 3 4 x 4 / 3 + x 7 / 3 ) dx = 3 x 4 / 3 12 7 x 7 / 3 + 3
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