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 fxd x Fb Fa =− Why is this true? Area(a,b+h) - Area(a,b) f(b) h so Area(a,b+h) - Area(a,b) h fb Problem 3 0 Find (8 2 ) xdx + Area under f(x) = 8 + 2x from a=0 to b=3 3 0 2 2 ) (3) (0) () 8 (3) 24 9 33 (0) 0 Area = (3) (0) 33 xd xF F Fx x x FF += =+ = = −= Area under f(x) = 8 + 2x from a=0 to b=3 12 Trapezoid with height 3, 8, 14 81 4 Area 3 33 22 hb b bb h === + + == = 8 14 03 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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Solution 20 2 0 23 5 (20) (0) 4 5 () 22 4 5 400 8000 1000 (20) 1000 4 3 666 yards .38 miles t td x F F tt Ft F −= =− = == Problem 2 0 Find 4 5 x ex x d x −− Solution 2 0 4 4 45 ( 2 ) ( 0 ) 4 5 4 (2) 8 20 27.3
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2.9 Class Notes - Fundamental theorem of calculus Why is...

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