# Calculus: One and Several Variables

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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-06 JWDD027-Salas-v1 November 25, 2006 15:39 288 SECTION 6.1 CHAPTER 6 SECTION 6.1 1. (a) ± 2 1 [( x +2) x 2 ] dx (b) ± 1 0 [( y ) ( y )] dy + ± 4 1 [( y ) ( y 2)] dy 2. (a) ± 0 4 ² ( 4 x ) x 2 ³ dx (b) ± 16 0 ´µ 1 4 y ( y ) · dy 3. (a) ± 2 0 ²( 2 x 2 ) ( x 3 dx (b) ± 8 0 ¸ ¹ y 1 / 3 º µ 1 2 y 1 / 2 » dy 4. (a) ± 1 0 ² x x 3 ³ dx (b) ± 1 0 ¼ y 1 / 3 y 2 ½ dy 5. (a) ± 4 0 ² (0) ( x dx + ± 6 4 [(0) ( x 6)] dx (b) ± 0 2 ² ( y +6) ( y 2 dy

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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-06 JWDD027-Salas-v1 November 25, 2006 15:39 SECTION 6.1 289 6. (a) ± 8 1 ² x 1 / 3 ³ x 2 3 ´µ dx (b) ± 2 1 (3 y +2) y 3 · dy 7. (a) ± 0 2 ²³ 8+ x 3 ´ ( x ) µ dx + ± 4 0 ²³ x 3 ´ ( x ) µ dx (b) ± 2 0 [( y ) ( y )] dy + ± 4 2 [( y ) (3 y 8)] dy 8. (a) ± 3 / 2 0 [2 x x ] dx + ± 3 3 / 2 [3 x ] dx (b) ± 3 0 ² y 1 2 y µ dy 9. (a) ± 5 4 ¶( 4+ x ) ( x dx (b) ± 3 3 (5) ( y 2 4 dy 10. (a) ± 2 0 [ x ( x )] dx (b) ± 0 2 [2 ( y )] dy + ± 2 0 [2 y ] dy
P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-06 JWDD027-Salas-v1 November 25, 2006 15:39 290 SECTION 6.1 11. (a) ± 3 1 [(2 x ) ( x 1)] dx + ± 5 3 [(9 x ) ( x 1)] dx (b) ± 4 2 ² ( y +1) ³ 1 2 y ´µ dy + ± 6 4 ² (9 y ) ³ 1 2 y ´µ dy 12. (a) ± 1 1 x 3 ( x 2 + x 1) · dx (b) ± 1 5 / 4 ²³ 1 2 + 1 2 ¸ 4 y +5 ´ ³ 1 2 1 2 ¸ 4 y ´µ dy + ± 1 1 ²³ 1 2 + 1 2 ¸ 4 y ´ y 1 / 3 µ dy 13. (a) ± 1 1 ¹º x 1 / 3 » ( x 2 + x 1 ) ¼ dx (b) ± 1 5 / 4 ²³ 1 2 + 1 2 ¸ 4 y ´ ³ 1 2 1 2 ¸ 4 y ´µ dy + ± 1 1 ²³ 1 2 + 1 2 ¸ 4 y ´ ( y 3 ) µ dy 14. (a) ± 3 1 ² ( x ³ x 1 3 ´µ dx + ± 5 3 ² (13 3 x ) ³ x 1 3 ´µ dx (b) ± 0 2 ²³ 13 y 3 ´ ( 3 y 1) µ dy + ± 4 0 ²³ 13 y 3 ´ ( y 1) µ dy 15. A = ± 3 0 ²³ 4 y y 2 4 ´ º y 4 » µ dy = ± 3 0 ³ 3 4 y 1 4 y 2 ´ dy = 3 8 y 2 1 12 y 3 · 3 0 = 9 8

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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-06 JWDD027-Salas-v1 November 25, 2006 15:39 SECTION 6.1 291 16. A = ± 2 1 ² (4 y 2 ) (2 y ) ³ dy = ± 2 1 (2 + y y 2 ) dy = ´ 2 y + y 2 2 y 3 3 µ 2 1 = 9 2 17. A =2 ± 2 0 ²( 12 2 y 2 ) ( y 2 dy ± 2 0 ( 12 3 y 2 ) dy ² 12 y y 3 ³ 2 0 = 2(16) = 32 2 4 6 8 10 12 x -2 -1 1 2 y 18. A = ± 1 0 ´ y 1 / 3 (2 y 2 y ) µ dy = ± 1 0 ( y 1 / 3 + y 2 y 2 ) dy = 3 4 y 4 / 3 + y 2 2 2 3 y 3 · 1 0 = 7 12 19. A = ± 0 2 ²( y 3 y ) ( y y 2 dy + ± 1 0 ²( y y 2 ) ( y 3 y dy = ± 0 2 ( y 3 + y 2 2 y ) dy + ± 1 0 ( 2 y y 2 y 3 ) dy = ´ 1 4 y 4 + 1 3 y 3 y 2 µ 0 2 + ´ y 2 1 3 y 3 1 4 y 4 µ 1 0 = 8 3 + 5 12 = 37 12 -6 -4 -2 1 x -2 2 y 20. A = ± 0 2 1 8 (2 y 3 + y 2 2 y ) 1 8 y 3 · dy + ± 1 0 1 8 y 3 1 8 (2 y 3 + y 2 2 y ) · dy = 1 8 y 4 4 + y 3 3 y 2 · 0 2 + 1 8 y 4 4 y 3 3 + y 2 · 1 0 = 37 96
P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-06 JWDD027-Salas-v1 November 25, 2006 15:39 292 SECTION 6.1 21. A = ± π/ 4 π/ 4 ² sec 2 x cos x ³ dx =2 ± π/ 4 0 ² sec 2 x cos x ³ dx = 2[tan x + sin x ] π/ 4 0 ² 1+ 2 / 2 ³

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## This note was uploaded on 04/30/2011 for the course MATH 1431 taught by Professor Any during the Spring '08 term at University of Houston.

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