INTRODUCTORY MATHEMATICAL ANALYSIS For Busiess, Ecoomics, ad the Life ad Social Scieces Chapter 4 Itegratio 0 Pearso Educatio, Ic.
Chapter 4: Itegratio Chapter Objectives To defie the differetial. To defie the ati-derivative ad the idefiite itegral. To evaluate costats of itegratio. To apply the formulas for u du, e du ad du. u To hadle more challegig itegratio problems. To evaluate simple defiite itegrals. To apply Fudametal Theorem of Itegral Calculus. 0 Pearso Educatio, Ic.
Chapter 4: Itegratio Chapter Objectives To use Trapezoidal rule or Simpso s rule. To use defiite itegral to fid the area of the regio. To fid the area of a regio bouded by two or more curves. To develop cocepts of cosumers surplus ad producers surplus. 0 Pearso Educatio, Ic.
Chapter 4: Itegratio 4.) 4.) 4.) 4.4) 4.6) 4.7) Differetials Chapter Outlie The Idefiite Itegral Itegratio with Iitial Coditios More Itegratio Formulas The Defiite Itegral The Fudametal Theorem of Itegral Calculus 0 Pearso Educatio, Ic.
f(x + dx) Q 4. Differetials secat lie The slope of the taget lie at (x, f(x)) is f(x) P dx x z dy x + dx y ( x) f dy dx taget lie (L) f whe 0, Δ ( x) f dy dx ( x+ dx) f( x) + y f( x) + dy f( x) + f ( x)dx 0 Pearso Educatio, Ic.
Ex Computig a Differetial The differetial of y, deoted dy or d(f(x)), is give by dy f ' ( x ) x dy f ' ( x )dx Fid the differetial of y x x + x 4 ad evaluate it whe x ad x 0.04. Solutio: The differetial is d dy x x + x 4 x x 4x+ dx ( ) ( ) x Whe x ad x 0.04, dy ( ( ) 4( ) + )( 0.04) 0. 08 0 Pearso Educatio, Ic.
Ex: ) Use differetials to estimate 7.97 whe. ) Give +, a) Fid the chages of y value whe, 0.0. b) Compare the true value of y. 0 Pearso Educatio, Ic.
f ( x+ dx) f( x) + y f( x) + dy f( x) + f ( x)dx Ex A govermetal health agecy examied the records of a group of idividuals who were hospitalized with a particular illess. It was foud that the total proportio P that are discharged at the ed of t days of hospitalizatio is give by 00 P P( t) ( ) 00 + t Use differetials to approximate the chage i the proportio discharged if t chages from 00 to 05. 0 Pearso Educatio, Ic.
Chapter 4: Itegratio Ex Solutio: We approximate P by dp, Ex 00 00 P dp P' ( t) 4 dt ( t) ( t) dt 00+ 00+ Fid dp dq if q 500 p. Solutio: dq dp p 500 p dp dq dq dp 500 p p 0 Pearso Educatio, Ic.
Ex: Give a) Evaluate b) Use differetials to estimate the value of f(0.98) 0 Pearso Educatio, Ic.
4. The Ifiite Itegral A atiderivative of a fuctio f is a fuctio F ( ) ( ) such that F ' x f x. I differetial otatio, Itegratio states that df f ( x)dx ( x) dx f( x) dx F( x) + C if oly F' ( x) f( x) df 0 Pearso Educatio, Ic.
Basic Itegratio Properties: 0 Pearso Educatio, Ic.
Ex - Fidig a Idefiite Itegral Fid 5 dx. 5dx 5x+ C Ex - Idefiite Itegral of a Costat Times a Fuctio Fid. 7xdx 7x 7x dx + C Ex: a. t dx t / dx / t / + C t + C b. 6x dx 6 x dx x 6 + C x + C 0 Pearso Educatio, Ic.
Ex - Idefiite Itegral of a Sum ad Differece Fid ) ( x x x + 4e )dx ) ( 4e x )dx ) 5x 4 dx 0 Pearso Educatio, Ic.
Fid 4) u + u du 5) ( x x x )dx 6) 4 x 5x + 5x x dx 0 Pearso Educatio, Ic.
Fid a. ( x )( 4x ) dx 8 b. x 4x dx 0 Pearso Educatio, Ic.
4. Itegratio with Iitial Coditios Use iitial coditios to fid the costat, C. If y is a fuctio of x such that y 8x 4 ad y() 5, fid y. Solutio: We fid the itegral, x y 4 + ( 8x 4) dx ( 8) 4x+ C 4x x C Usig the coditio, The equatio is 5 4 ( ) 4( ) + C C y 4x 4x 0 Pearso Educatio, Ic.
Ex - Icome ad Educatio For a particular urba group, sociologists studied the curret average yearly icome y (i dollars) that a perso ca expect to receive with x years of educatio before seekig regular employmet. They estimated that the rate at which icome chages with respect to educatio is give by dy / 00x 4 x 6 dx where y 8,70 whe x 9. Fid y. 0 Pearso Educatio, Ic.
Solutio: We have / 5 / y 00x dx 40x + C Whe x 9, 8,70 40 C 9,000 Therefore, y 40x 5 / ( 9) 5 / + C + 9,000 0 Pearso Educatio, Ic.
Ex: dr/dq is the margial-reveue fuctio. Fid the demad fuctio for the followig: dr dq ( ) + 5000 q q 0 Pearso Educatio, Ic.
Ex: Fid the total cost fuctio where the fixed cost is $000 ad the margial cost fuctio is dc dq q+ 75 0 Pearso Educatio, Ic.
4.4 More Itegratio Formulas Power Rule for Itegratio + u u dx + C + if Itegratig Natural Expoetial Fuctios u u e du e + C Itegrals Ivolvig Logarithmic Fuctios 0 Pearso Educatio, Ic. dx x l x + C for x 0
Basic Itegratio Formulas 0 Pearso Educatio, Ic.
Ex Fid the itegral of 0 ( x ) dx a. + Let u x+, the du dx ( x+ ) ( ) 0 u x dx ( u) 0 + du + C + C b. ( x 7) dx x + Let u x + 7 du x dx 4 4 u x + 7 x + 4 + 4 ( ) ( ) ( ) x 7 dx u du + C C 0 Pearso Educatio, Ic.
Ex Applyig the Power Rule for Itegratio Fid a. 6ydy b. x + x ( x + x + ) dx 4 4 7 0 Pearso Educatio, Ic.
Ex - Itegrals Ivolvig Expoetial Fuctios Fid a. xe x dx b. ( ) x + x x e dx + 0 Pearso Educatio, Ic.
Fid a. ( 7) x dx. x + x b. Give y ad iitial valuey( ) x + 6 fidy. 0, 0 Pearso Educatio, Ic.
.5 Summatio Notatio DEFINITION The sum of the umbers a i, with i successively takig o the values m through is deoted as i am + am+ + am + +... i m a + a 0 Pearso Educatio, Ic.
Evaluate the give sums. a. 7 ( 5 ) b. 7 ( 5 ) [ 5( ) ] + [ 5( 4) ] + [ 5( 5) ] + [ 5( 6) ] + [ 5( 7) ] 6 ( j + ) j + 8+ + 8+ 5 6 j ( j + ) ( + ) + ( + ) + ( + ) + ( 4 + ) + ( 5 + ) + ( 6 + ) 0 Pearso Educatio, Ic. + 5+ 0+ 7+ 6+ 7 97
.5 Summatio Notatio Cosider the sum of the first itegers: + + + + i i ( + ) Accordig to mathematical leged, the famous mathematicia Karl Friedrich Gauss discovered this formula whe he was about seve years old usig the followig argumet. 0 Pearso Educatio, Ic.
S + + +... + +S + - + - +... + S ( + ) + ( + ) + ( + ) +... + ( + ) S ( + ) i i ( + ) 0 Pearso Educatio, Ic.
a. i e. i c c b. i i ( + ) c. d. i i i i ( + )( ) + 6 ( ) + 4 0 Pearso Educatio, Ic.
0 Pearso Educatio, Ic. m i i m i i a c ca a. ( ) ± ± m i i m i i m i i i b a b a b.
Ex: Fid the sum a. 6 ( ) d. 00 k 5 k b. 0 k k c. 7 ( ) 0 Pearso Educatio, Ic.
Evaluate the give sums. a. 00 j 0 4 d. 60 k 0 ( k + ) b. 00 ( 4k+ ) k 9 c. 60 k 0 9k 0 Pearso Educatio, Ic.
4.6 The Defiite Itegral For area uder the graph from limit a b, b a f ( x)dx x is called the variable of itegratio ad f (x) is the itegrad. 0 Pearso Educatio, Ic.
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f(x) x + b/w 0 to, subdivided to four regio Right had ed poits Left had ed poits S S 0 Pearso Educatio, Ic.
Ex - Usig Right-Had Edpoits fid the area Fid the area of the regio i the first quadrat bouded by f(x) 4 x ad the lies x 0 ad y 0. Sice the legth of [0, ] is, x /. 0 Pearso Educatio, Ic.
0 Pearso Educatio, Ic. Ex - Computig a Area by Usig Right-Had Edpoits Summig the areas, we get We take the limit of S as : Hece, the area of the regio is 6/. ( )( ) ( )( ) + + + + 4 8 6 8 8 4 k f x k f S k k ( )( ) 6 8 8 4 8 lim lim + + S
Ex - Itegratig a Fuctio over a Iterval Itegrate f (x) x 5 from x 0 to x. Solutio: S k ( + ) 9 9 f k 5 + 5 0 ( x 5) dx lim S lim 9 + 5 9 0 Pearso Educatio, Ic.
Sketch the regio i the first quadrat that is bouded by the give curves. Determie the exact area of the regio by cosiderig the limit of as. Use the right-had edpoit of each subiterval.., y 0, x, x. 9, y 0, x 0, x 0 Pearso Educatio, Ic.
Fid S for the give fuctio. Use the right-had side edpoits. 4 ; 0, x 0 Pearso Educatio, Ic.
Ex: Fid where 4 x 5x 0 if if without the use of limits 0 if x x< x < 0 Pearso Educatio, Ic.
Chapter 4: Itegratio 4.7 The Fudametal Theorem of Itegral Calculus Fudametal Theorem of Itegral Calculus If f is cotiuous o the iterval [a, b] ad F is ay atiderivative of f o [a, b], the b a f ( x) dx F( b) F( a) Properties of the Defiite Itegral If a > b, the If limits are equal (a b), 0 Pearso Educatio, Ic. b a ( x) dx f( x) a f b b a f dx ( x) dx 0
Properties of the Defiite Itegral b a ( ). S f xdx is the area bouded by the graph f(x). 0 Pearso Educatio, Ic.
.. 4. 5. b ( x) dx k f( x) dx where k is a costat. kf a b b a [ f( x) ± g( x) ] dx f( x) dx g( x) ± a b ( x) dx f( t) f a c b a dt ( x) dx f( x) dx f( x) f + a b a b a c b dx b a dx 0 Pearso Educatio, Ic.
0 Pearso Educatio, Ic. Ex - Applyig the Fudametal Theorem Fid ( ). 6 + dx x x ( ) ( ) ( ) ( ) ( ) 48 6 6 6 6 + + + + x x x dx x x
Ex - Evaluatig Defiite Itegrals Fid [ ] 6x 4x 5 a. + dx b. e t dt 0 c. xdx 0 Pearso Educatio, Ic.
d. dx x x e. dx 4+ x 0 5 f. ( e+ e) dx 0 Pearso Educatio, Ic.
A maufacturer s margial-cost fuctio is 0.004 0.5 +50 If c is i dollars, determie the cost ivolved to icrease productio from 90 to 80 uits. 0 Pearso Educatio, Ic.