ISM_T11_C06_A

# ISM_T11_C06_A - CHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS 6.1 VOLUMES BY SLICING AND ROTATION ABOUT AN AXIS 1(a A 1(radius and radius 1 c x A(x

This preview shows pages 1–3. Sign up to view the full content.

CHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS 6.1 VOLUMES BY SLICING AND ROTATION ABOUT AN AXIS 1. (a) A (radius) and radius 1 x A(x) 1 x œœ ± Ê œ ± 11 ## # È ab (b) A width height, width height 2 1 A(x) 4 1 x œ ± Ê œ ± È # # (c) A (side) and diagonal 2(side) A ; diagonal 2 1 A(x) 2 1 x Ê œ œ ± Ê œ ± # # # # È È (diagonal) # (d) A (side) and side 2 1 A(x) 3 1 x ± Ê œ ± È 3 4 # È È 2. (a) A (radius) and radius x A(x) x Ê œ # È (b) A width height, width height 2 x A(x) 4x œ Ê œ È (c) A (side) and diagonal 2(side) A ; diagonal 2 x A(x) 2x Ê œ œ Ê œ # # È È (diagonal) # (d) A 2 x A(x) 3x Ê œ È 3 4 # È È 3. A(x) 2x (see Exercise 1c); a 0, b 4; œ œ œ (diagonal) xx # # ±± ˆ‰ ÈÈ V A(x) dx 2x dx x 16 œ œ '' a0 b4 cd # % ! 4. A(x) 1 2x x ; a 1, b 1; œ œ ± ² œ ± œ 1 (diameter) 44 4 2x x 2 1x # # c d ± #% 1 V A(x) dx 1 2x x dx x x 2 1 ± ² œ ± ² œ ± ² œ a1 b1 1 ’“ \$ " ±" " 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; ± ± ± ± œ ± œ ± œ ± œ # # # Š È Š‹ V A(x) dx 4 1 x dx 4 x 8 1 ± œ ± œ ± œ # " ±" " x1 6 33 3 \$ 6. A(x) 2 1 x (see Exercise 1c); a 1, b 1; œ œ ± œ ± œ (diagonal) 21x # # # ±±± ± ± # Š È V A(x) dx 2 1 x dx 2 x 4 1 ± œ ± œ ± œ # " ±" " x8 3 \$ 7. (a) STEP 1) A(x) (side) (side) sin 2 sin x 2 sin x sin 3 sin x œ "" †† 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 œ ± œ ² œ b È È 1 1 ! (b) STEP 1) A(x) (side) 2 sin x 2 sin x 4 sin x œ # STEP 2) a 0, b 1 STEP 3) V A(x) dx 4 sin x dx 4 cos x 8 œ ± œ b 1 1 ! 8. (a) STEP 1) A(x) (sec x tan x) sec x tan x 2 sec x tan x ± œ ² ± 1 (diameter) 4 # # sec x sec x 1 2 œ² ± ± 1 4c o s x sin x ± # STEP 2) a , b œ± œ STEP 3) V A(x) dx 2 sec x 1 dx 2 tan x x 2 ± ± œ ± ² ± a3 b3 Î Î 1 1 1 1 o s x 4 c o s x 2 sin x ± # " Î\$ ±Î \$ #

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
362 Chapter 6 Applications of Definite Integrals 23 2 2 43 œ± ² ± ± ± ² ² ± 11 1 1 1 3 4 3 2 ’“ Š ÈÈ È Š‹ Š "" ˆ‰ ## (b) STEP 1) A(x) (edge) (sec x tan x) 2 sec x 1 2 œœ ± œ ± ± # sin x cos x # STEP 2) a , b œ 33 STEP 3) V A(x) dx 2 sec x 1 dx 2 2 3 4 3 ± ± œ ± œ ± '' a3 b3 Î Î 1 1 # 2 sin x 2 cos x 3 3 # 9. A(y) (diameter) 5y 0 y ; ± œ 1 44 4 5 % # È c 0, d 2; V A(y) dy y dy œ œ c0 d2 5 4 1 % 208 ± œ 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; ± ± ± ± œ ± œ ± œ ± œ " # # # ± ‘ˆ È V A(y) dy 2 1 y dy 2 y 4 1 ± œ ± œ ± œ 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 œ œ œ 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 Êœ # 12. 1) The solid and the cone have the same altitude of 12.
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 10/13/2011 for the course MATHEMATIC 103 taught by Professor Thommas during the Spring '11 term at LCC Intl University.

### Page1 / 10

ISM_T11_C06_A - CHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS 6.1 VOLUMES BY SLICING AND ROTATION ABOUT AN AXIS 1(a A 1(radius and radius 1 c x A(x

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online