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EXERCISES 5.2

The Definita Integral and Integration

Because on Tuesday the area of the wound was 2 cm2, we know that A= 2

when t = 1. Substituting these values in (10) we obtain

2=1+C

C=l

Therefore from (10),

3 A=--+l

t+2

(a) On Monday, t = 0. Let A0 be the value of A when t = 0. From (11),

Ao = l + 1

=i Thus on Monday the area of the wound was 2.5 cm2•

(b) On Friday, t = 4. Let A4 be the value of A when t = 4. From (11),

A4 =i + 1

=t Hence on Friday the anticipated area of the wound is 1.5 cm2•

In Exercises 1 through 52, perform the antid!fferentiation. 29. Jcos x(2 + sin x)5 dx 30 f 4 sin x dx

I. f ../1 -'- 4y dy l. J {/3x - 4 dx

3. J {/.6 - 2x dx 4. J$r+t dr

I ' (1 + cos x)2

~31. f J 1 + 3~ ~~ 32. IF~ 5. J x../x2

- 9 dx 6. J3x../4- x2 dx IJ3. J 2 sin x {/1 + cos x dx 34. J sin 2x ,J,-2 ---co-s-::2-x dx

7. J x 2(x3 - 1)1° dx 8. J x(2x2 + 1)6 dx 35. J cos2 t sin t dt 36. J sin3 (J cos (J d(J

9. J Sx {1(9 - 4x2)

2 dx f xdx 10. (x2 + l)l

37. J<tan 2.x: + cot 2x)2 dx 38. · f ! cos !x dx ../sin !x

. f y3 dy ll. f sds 11. (1 - 2y4)5 ../3s2 + 1

13. J (x2 - 4x + 4f'3 dx /t4. J x4 ../3x5

- 5 dx

39. f cos 3x dx 40. f sec2 ~ .[t dt ../1- 2 sin 3x y;

41. f (x2 + 2x) dx . . '\.. ../x3 + 3x2 +·I (

/15. J x.Jx + 2 dx 16. f __.!_!!__ ..;t+3

J 2rdr 18. J x3(2- x2)12 dx ,

17' (1- rf

i it. J ../3 - 2x x2 dx 20. J (x3 + 3)1'4x5 dx

42. J x(x2 + 1).J4- 2x2 - x4 dx

f x(3x2 +1)dx J ~ 2 43. (3x4 + 2x2 + 1)2 44. v3 + s(s + 1) ds

45. f (y + 3) dy 46. f(2t2 + l)li3t3 dt (3- y)213

J 21. f cos 49 d9 22. J sin !x dx

' 23. J 6x2 sin x 3 dx 24. J tr cos 4t2 dt

~ 25. J sec2 Sx dx 26. J csc2 29 d9

,., '¥1. J y esc 3yl cot 3yl dy 28. J r2 sec2 rl dr

f (r1i3 + 2f dr f ( 1)3'2(r2- 1)

47. .a 48. t +- - 2- dt ""'2 t t

f x3

dx 50. f X3

dx 49. (xl + 4)3/2 ../t - 2xz

51. J sin' x sin(cos x) dx lsl. J sec x tan x cos(sec x) dx

i

5.3 ... EOIMI

RECTiu.£1

dx

6.3 DifflnMial EqaltiGIII ..id Rlctilinlar Metion 313

J (2x + 1)3 dx by two methods: (a) Expand (2x + 1)3 by the binomial theorem; (b) let u = 2x + 1. (c) Explain the difference in appearance of the answers obtained in (a) and (b). Evaluate J x(x2 + 2)2 dx by two D¥'thods: .(a) Expand ~ + 2)2 and multiply the result by x; (b) let u = x2 + 2. (c) Explain the difference in appearance of the answers obtained in (a) and (b).

f (../X- 1)2 Evaluate ../X dx by two methods: (a) Expand

(Vx- 1)2 and multiply the Fe$Ult by x- 112; (b) let u = ../X - 1. (c) Explain the difference in appearance of .the answers obtained in (a) and (b~

Evaluate J ~ x 2 dx by two methods: (a) Let u = x - 1; (b)letv=~. Evaluate J 2 sin x cos x dx by three methods: (a) Let u = sin x; (b) let v = cos x; (c) use the identity 2 sin x cos x = sin 2x. (d) Explain the difference in appear• ance of the answers obtained in (a), (b), and (c).

Evaluate J csc2 x cot x dx by two methods: (a) Let u = cot x;

(b) let v =esc x. (c) Explain the difference in appearance of the answers obtained in (a) and (b). · The marginal cost function for a particular article of mer­c:handise is given by C'(-.x) = 3(5x + 4)- 1

'2

• H the overhead cost is $10, find the total cost function. For a certain commodity the marginal cost function is given by C'(x) = 3..)2x + 4. H the overhead cost is zero, find the total cost function. H x units are demanded when p dollars is the price per unit, find an equation involving p and x (the demand equation) of a commodity for which the marginal revenue function is given by R'(x) = 4 + 10(x + 5)- 2

•

Cil. The marginal revenue function for a particular article of mer­chandise is given by R'(x) = ab(x + b)- 2 - c. Find (a) the to­tal revenue function and (b) an equation involving p and x (the demand equation) where x units are demanded when p dollars is the price per unit.

63. If q. coulombs is the charge of electricity received by a con­denser from an electric current of i amperes at t seconds then

i = ~- If i = 5 sin 60t and q = 0 when t = !1t, find the

greatC$t positive charge on the condenser. 64. Do Exercise 63 if i = 4 cos 120t and q = 0 when t = 0. 65. The cost of a certain piece of machinery is $700, and its

value is depreciating with time according to the formula dY dt = -SOO(t + 1)-2, where Y dollars is its value tyears after

its purchase. What is its value 3 years after its purchase? 66. The volume of a balloon is increasing according to the for-

dY · '· mula dt = .Jif- 1 + jt, where Y cubic centimeters is the vol-

ume of the balloon at t seconds. H Y = 33 when t = 3, find (a) a formula for Yin terms oft; (b) the volume of the balloon at 8 sec.

67 •. For the first 10 days in December a plant cell grew in such a way that t days after December 1 the volume of the cell was increasing at a rate of(12- t)- 2 cubic micrometers pet: day. If on December 3 the volume of the cell was 3 Jllll3, w­was the volume on December 8? · •

fi8. The volume of water in a tank is Y cubic meters when the depth of the water is h meters. If the rate of change or y witJi

respect to h is given by ~~ • K(2h + 3)2, find the volUme of

water in the tank, when, tho d~th is 3 m.

5.3 DIFFERENTIAL. An equation containing derivatives is ailled a ditfereatial eqatioa. Some simple differential eq~tions are ' EQUATIONS AND

RECTILINEAR MOTION ~~ = 2x a nlUl'"' (l)

dy 2x2 dx = 3y3

d2y dx2 = 4x + 3

(2)

(3)

The order of a differential equation is the order of the derivative of highest order that appears in the equation. Therefore (l) and (2) are first-order differen­tial .equations and (3) is of the second order. The simplest type of differential

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