2.9 Class Notes - Fundamental theorem of calculus Why is...

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Fundamental theorem of calculus Given any antiderivative F the area under the curve between a and b is given by the definite integral ( ) ( ) ( ) b a f x dx F b F a = Why is this true? Area(a,b+h) - Area(a,b) f(b) h so Area(a,b+h) - Area(a,b) ( ) h f b Problem 3 0 Find (8 2 ) x dx + Area under f(x) = 8 + 2x from a=0 to b=3 3 0 2 (8 2 ) (3) (0) ( ) 8 (3) 24 9 33 (0) 0 Area = (3) (0) 33 x dx F F F x x x F F F F + = = + = + = = = Area under f(x) = 8 + 2x from a=0 to b=3 1 2 1 2 Trapezoid with height 3, 8, 14 8 14 Area 3 33 2 2 h b b b b h = = = + + = = = 8 14 0 3 Porsche 928, p. 415 The velocity at t seconds is 10t – t 2 /4 for t 20 seconds How far does the car move during this time? 1 mile per hour ½ yard per second 0 20 40 60 80 100 0 5 10 15 20
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