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 = " 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 . 919405 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 + µ π 2 + ( n 1) π n ³ 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 = 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. 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 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 ) 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) Z x x 2 dx = p x 2 + C (b) Z ( 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 µ 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 h p x 3 +5 i = 3 x 2 2 x 3 so Z 3 x 2 2 x 3 dx = p 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 Z 3 x 2 ( x 2 2 dx = x x 2 + C 5. d dx £ sin ( 2 x = cos (2 x ) x so Z cos (2 x ) x dx = sin ( 2 x ) + C 6. d dx [sin x x cos x ]= x sin x so Z x sin xdx = 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 Z x 3 dx = 1 4 x 2 + C (b) u 4 / 4 u 2 +7 u + C 10.
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Calculus: Early Transcendentals, by Anton, 7th Edition,ch06...

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