6.1 Rebanadas - CHAPTER 6 APPLICATIONS OF DEFINITE...

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CHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS 6.1 VOLUMES BY SLICINGAND ROTATION ABOUT AN AXIS 1. (a) A (radius) and radius 1 x A(x) 1 x oo ± Ê o ± 11 ## # È ab (b) A width height, width height 2 1 A(x) 4 1 x o ± Ê o ± È # # (c) A (side) and diagonal 2(side) A ; diagonal 2 1 A(x) 2 1 x Ê o o ± Ê o # # # # È È (diagonal) (d) A (side) and side 2 1 A(x) 3 1 x ± Ê o ± È 3 4 # È È 2. (a) A (radius) and radius x A(x) x Ê o # È (b) A width height, width height 2 x A(x) 4x o Ê o È (c) A (side) and diagonal 2(side) A ; diagonal 2 x A(x) 2x Ê o o Ê o # # È È (diagonal) (d) A 2 x A(x) 3x Ê o È 3 4 # È È 3. A(x) 2x (see Exercise 1c); a 0, b 4; o o o (diagonal) xx ±± ˆ‰ ÈÈ V A(x) dx 2x dx x 16 o o '' a0 b4 cd # % ! 4. A(x) 1 2x x ; a 1, b 1; o o ± ² o ± o 1 (diameter) 44 4 2x x 2 1x c d ± #% 1 V A(x) dx 1 2x x dx x x 2 1 ± ² o ± ² o ± ² o a1 b1 1 ’“ $ " ± " " 2x 2 1 6 35 3 5 1 5 1 5. A(x) (edge) 1 x 1 x 2 1 x 4 1 x ; a 1, b 1; ± ± ± ± o ± o ± o ± o # # # Š È Š V A(x) dx 4 1 4 x 8 1 ± o ± o ± o # " ± " " x1 6 33 3 6. A(x) 2 1 x (see Exercise 1c); a 1, b 1; o o ± o ± o (diagonal) 21x # ±±± ± ± # Š È Š V A(x) dx 2 1 2 x 4 1 ± o ± o ± o # " ± " " x8 3 7. (a) STEP 1) A(x) (side) (side) sin 2 sin x 2 sin x sin 3 sin x o "" †† Š Š È STEP 2) a 0, b 1 STEP 3) V A(x) dx 3 sin x dx 3 cos x 3(1 1) 2 3 o ± o ² o b È È 1 ! (b) STEP 1) A(x) (side) 2 sin x 2 sin x 4 sin x o # Š Š STEP 2) a 0, b 1 STEP 3) V A(x) dx 4 sin x dx 4 cos x 8 o ± o b 1 ! 8. (a) STEP 1) A(x) (sec x tan x) sec x tan x 2 sec x tan x ± o ² ± 1 (diameter) 4 # sec x sec x 1 2 o ² ± ± 1 4c o s x sin x ± STEP 2) a , b o ± o STEP 3) V A(x) dx 2 sec x 1 dx 2 tan x x 2 ± ± o ± ² ± a3 b3 1 1 o s x 4 c o s x 2 sin x ± # " Î$ ± Î $
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362 Chapter 6 Applications of Definite Integrals 2 3 2 2 3 2 4 3 o ± ² ± ± ± ² ² ± o ± 11 1 1 1 43 3 4 3 2 ’“ Š ÈÈ È Š Š Š "" ˆ‰ (b) STEP 1) A(x) (edge) (sec x tan x) 2 sec x 1 2 oo ± o ± ± ## # sin x cos x STEP 2) a , b o ± o 33 STEP 3) V A(x) dx 2 sec x 1 dx 2 2 3 4 3 ± ± o ± o ± '' a3 b3 Š # 2 sin x 2 cos x 3 3 9. A(y) (diameter) 5y 0 y ; ± o 1 44 4 5 % # Š È c 0, d 2; V A(y) dy y dy o o c0 d2 5 4 1 % 2 0 8 ± o Š ab 5 45 4 y # ! & 1 10. A(y) (leg)(leg) 1 y 1 y 2 1 y 2 1 y ; c 1, d 1; ± ± ± ± o ± o ± o " # # # ± ‘ˆ È V A(y) dy 2 1 2 y 4 1 ± o ± o ± o c1 d1 # " ± " " y 3 8 11. (a) It follows from Cavalieri's Principle that the volume of a column is the same as the volume of a right prism with a square base of side length s and altitude h. Thus, STEP 1) A(x) (side length) s ; STEP 2) a 0, b h; STEP 3) V A(x) dx s dx s h o o o a0 bh (b) From Cavalieri's Principle we conclude that the volume of the column is the same as the volume of the prism described above, regardless of the number of turns V s h Êo # 12. 1) The solid and the cone have the same altitude of 12. 2) The cross sections of the solid are disks of diameter x . If we place the vertex of the cone at the ± o xx origin of the coordinate system and make its axis of symmetry coincide with the x-axis then the cone's cross sections will be circular disks of diameter (see accompanying figure).
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6.1 Rebanadas - CHAPTER 6 APPLICATIONS OF DEFINITE...

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