20 23 Evaluate the integral by interpreting it in terms of areas. (1 x ) dx. y 2 2
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1 SECTION 5. THE DEFINITE INTEGRAL 5. THE DEFINITE INTEGRAL A Clck here for swers. S Clck here for solutos. 7 Use the Mdpot Rule wth the gve vlue of to pproxmte the tegrl. Roud the swer to four decml plces. Evlute the tegrl by terpretg t terms of res.... y y x dx. y x dx y 5 x dx, 5 dx, x 7 y. s x dx,. y t x dx, y x dx.. y x 5 dx y 5. ssx dx, 5. y sec x dx, 7 Wrte the gve sum or dfferece s sgle tegrl the form. x b 7. y x l x dx,. y y y 8 9 Express the lmt s defte tegrl o the gve tervl. 5. y 8 5 y 5 8. lm x* 5x* x, l,. y y 7 9. lm sx* x, l, 7. y 5 y y 5 9 Use the form of the defto of the tegrl gve Theorem to evlute the tegrl. y b. c dx. y. x dx. y. x b dx 5. y. t t dt 7. y 7 x dx y 5 x x dx y x x dx y b Px Qx R dx 8. If d, fd x x8.7 x If x f t dt, x f t dt, d f t dt, fd. Use Property 8 to estmte the vlue of the tegrl. y x f t dt x. x x dx.. y s x dx y cos x dx y b 8. x x dx 9. y 5 t t dt Copyrght, Cegge Lerg. All rghts reserved.
2 SECTION 5. THE DEFINITE INTEGRAL 5. ANSWERS E Clck here for exercses. S Clck here for solutos x 5x dx 9. xdx. b c b b 7. P b + Q + R b 8. b +b Copyrght, Cegge Lerg. All rghts reserved f x dx 8 5. f x dx. f x dx 7 7. f x dx x +x dx 9. / cos x dx /. +x dx
3 SECTION 5. THE DEFINITE INTEGRAL 5. SOLUTIONS E Clck here for exercses.. The wdth of the tervls s x =5 /5 =so the prtto pots re,,,,, 5 d the mdpots re.5,.5,.5,.5,.5. The Mdpot Rule gves x dx 5 f x x = = 5.5. The wdth of the tervl x = / =.5 so the prtto pots re.,.5,.,.5,. d the mdpots re.5,.75,.5,.75. x 7 dx f x x = x = /5 =, so the edpots re,,,, 8,d, d the mdpots re,, 5, 7,d9. The Mdpot Rule gves s xdx 5 f x x = s +s +s 5+s 7+s 9.. x = / =, so the edpots re,,,,, 5,d, d the mdpots re,, 5, 7, 9. The Mdpot Rule gves d sec x/ dx f x x =.979 sec +sec +sec5 +sec 7 9 +sec +sec,. x = /=. so the prtto pots re.,.,...,. d the mdpots re.5,.5,...,.95. +x dx f x x = x = / =.5, so the edpots re,.5,,.5, d, d the mdpots re.5,.75,.5,d.75. The Mdpot Rule gves x l xdx f x x f x =x l x =.5f.5 + f.75 + f.5 + f O,, lm x 5x x = x 5x dx. Copyrght, Cegge Lerg. All rghts reserved.. x = = so the prtto pots re,,,, d the mdpots re,, 5, 7. The Mdpot Rule gves / t x dx t +t.5 +t 5 +t7 9. O,, lm b. b cdx x x = xdx. c b c =b c b c
4 SECTION 5. THE DEFINITE INTEGRAL Copyrght, Cegge Lerg. All rghts reserved x dx =9 8 = 9. x dx = 9 +9 =5 +x x dx = +8+= 8 x + b dx + b + b + + b + + b = + b + 5. x x dx =9 5 +9=9.5. t t + dt = + = b 7. Px + Qx + R dx b P b b P + b b P + Q + b + R +P+ Q b + P + Q + R +P+ Q b + P + Q + R b
5 SECTION 5. THE DEFINITE INTEGRAL 5 8. b + + P P b + +P+ Q P+ Q b + + P + Q + R b + + b + b = P + P+ Q b + P + Q + R b + P + Q + R b b = P b + b + b b + + b b +Q b + + b + R b b = P b + Q + R b b x +x dx b b b b + + b b + + b + b + +b = =.5. + x dx cbeterpretedsthereuderthegrph of f x =+x betwee x =d x =. Thssequl tothereoftherectgleplusthereofthetrgle,so + x dx = A = + =... Or: Use the formul for the re of trpezod: A = + 7 =. x dx c be terpreted s A A,whereA d A re the res of the trgles show. Thus, x dx = =. x dx c be terpreted s the re of the mddle trgle mus the res of the outsde oes, so x dx = =. Copyrght, Cegge Lerg. All rghts reserved. 9. = b +b t t + dt x 5 dx c be terpreted s the re uder the grph of the fucto f x = x 5 betwee x =d x =. Ths s equl to the sum of the res of the two trgles, so x 5 dx = =.. f x dx + f x dx + f x dx = f x dx + f x dx = f x dx
6 SECTION 5. THE DEFINITE INTEGRAL 8 5. f x dx + f x dx 5 = f x dx + 8 f x dx = 8 f x dx 5. f x dx 7 f x dx = 7 f x dx+ f x dx 7 f x dx = f x dx f x dx f x dx + f x dx 5 = f x dx + f x dx f x dx + f x dx 5 = f x dx 8. f x dx + 8 f x dx = 8 f x dx 5 f x dx +.5 =.7 f x dx =.8 9. f t dt + f t dt + f t dt = f t dt + f t dt += f t dt = = 9. If f x =x +x, x,thef x =x += whe x =, df =. At the edpots, f =, f =. Thus the bsolute mmum s m = d the bsolute mxmum s M =.Thus x +x dx or x +x dx 9.. If x, the cos x,so / cos x dx / or / cos x dx. /. For x, x d +x,so +x dx or +x dx. Copyrght, Cegge Lerg. All rghts reserved.
19 22 Evaluate the integral by interpreting it in terms of areas. (1 x ) dx. y Write the given sum or difference as a single integral in
SECTION. THE DEFINITE INTEGRAL. THE DEFINITE INTEGRAL A Clck here for swers. S Clck here for solutos. Use the Mdpot Rule wth the gve vlue of to pproxmte the tegrl. Roud the swer to four decml plces. 9
Area and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]
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this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]
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