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FDWKCalcSM_ch5 - 234 Section 5.1 Chapter 5 The Definite...

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234 Section 5.1 Chapter 5 The Definite Integral Section 5.1 Estimating with Finite Sums (pp. 263-273) Exploration 1 Which RAM is the Biggest? 1. LRAM > MRAM > RRAM 2. MRAM > RRAM > LRAM 3. RRAM > MRAM > LRAM, because the heights of the rectangles increase as you move toward the right under an increasing function. 4. LRAM > MRAM > RRAM, because the heights of the rectangles decrease as you move toward the right under a decreasing function. Quick Review 5.1 1. 80 5 400 mph hr mi i = 2. 48 3 144 mph hr mi i = 3. 10 10 100 ft/sec ft/sec 2 i sec = 100 ft/sec mi ft h mph i i 1 5280 3600 1 68 18 sec . 4. 300 000 3600 1 24 1 365 , / sec sec km hr hr day days i i i 1 1 yr yr i × 9 46 10 12 . km 5. ( )( ) ( )( ) 6 3 5 2 18 10 28 mph h mph h mi mi mi + = + = 6. 20 1200 gal/min 1 h 60 min 1h gal i i = 7. ( ( . )( − ° + ° = − ° 1 1 5 6 3 C/h)(12 h) C h) C 8. 300 1 24 1 25 920 ft /sec 3600 sec h h 1 day day 3 i i i = , ,000 ft 3 9. 350 50 17 500 people/mi mi people 2 2 i = , 10. 70 3600 1 1 0 7 176 400 times/sec h h times i i i sec . , = Section 5.1 Exercises 1. Since v ( t ) = 5 is a strait line, compute the area under the curve. x t v t = = = ( ) ( ) ( )( ) 4 5 20 2. Since v t t ( ) = + 2 1 creates a trapezoid with the x-axis, compute the area of the curve under the trapezoid. A h a b a t v b t v = + = = = = + = = = = = 2 0 0 2 0 1 1 4 4 2 4 ( ) ( ) ( ) ( ) ( ) + = = = + = 1 9 4 4 2 9 1 20 h A ( ) 3. Each rectangle has base 1. The height of each rectangle is Found by using the points t = ( . , . , . , . ) 0 5 1 5 2 5 3 5 in the equation v t t ( ) . 2 1 + The area under the curve is approximately 1 5 4 13 4 29 4 53 4 25 + + + = , so the particle is close to x = 25. 4. Each rectangle has base 1. The height of each rectangle is found by using the points y = ( . , . , . , . , . ) 0 5 1 5 2 5 3 5 4 5 in the equation v t t ( ) . = + 2 1 The area under the curve is approximately 1 5 4 13 4 29 4 53 4 85 4 46 25 + + + + = . , so the particle is close to x = 46 25 . .
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Section 5.1 235 5. (a) y 2 x 2 R (b) y x 2 2 Δ x = + 1 2 0 1 2 2 1 2 1 2 2 LRAM: [2(0) ( ) ] + 2 2 1 2 2 1 1 1 2 [ ( ) ( ) ] + = = 2 3 2 3 2 1 2 5 4 1 2 .25 6. (a) y x 2 2 RRAM: 2 1 2 1 2 1 2 2 2 + [ ( ) ( ) ] 1 1 1 2 2 3 2 3 2 2 2 + + = = 1 2 2 2 2 1 2 5 4 1 25 2 [ ( ) ( ) ] . (b) y x 2 2 MRAM: 2 1 4 1 4 1 2 2 3 2 + 4 3 4 1 2 2 5 4 2 + + 5 4 1 2 2 7 4 7 4 2 = = 2 1 2 11 8 1 375 .
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