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4-4 THE FUNDAMENTAL THEOREM OF CALCULUS MS. BATTAGLIA – AP CALCULUS
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THE FUNDAMENTAL THEOREM OF CALCULUS If a function f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then
![THE FUNDAMENTAL THEOREM OF CALCULUS If a function f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then](http://images.slideplayer.com./33/8262703/slides/slide_2.jpg "THE FUNDAMENTAL THEOREM OF CALCULUS If a function f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then")
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EVALUATING A DEFINITE INTEGRAL Evaluate each definite integral.
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A DEFINITE INTEGRAL INVOLVING ABSOLUTE VALUE
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USING THE FUNDAMENTAL THEOREM TO FIND AREA Find the area bounded by the graph of y = 2x 2 – 3x + 2, the x-axis, and the vertical lines x = 0 and x = 2.
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THE MEAN VALUE THEOREM FOR INTEGRALS If f is continuous on the closed interval [a,b], then there exists a number c in the closed interval [a,b] such that
![THE MEAN VALUE THEOREM FOR INTEGRALS If f is continuous on the closed interval [a,b], then there exists a number c in the closed interval [a,b] such that](http://images.slideplayer.com./33/8262703/slides/slide_6.jpg "THE MEAN VALUE THEOREM FOR INTEGRALS If f is continuous on the closed interval [a,b], then there exists a number c in the closed interval [a,b] such that")
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AVERAGE VALUE OF A FUNCTION ON AN INTERVAL If f is integrable on the closed interval [a,b], then the average value of f on the interval is
![AVERAGE VALUE OF A FUNCTION ON AN INTERVAL If f is integrable on the closed interval [a,b], then the average value of f on the interval is](http://images.slideplayer.com./33/8262703/slides/slide_7.jpg "AVERAGE VALUE OF A FUNCTION ON AN INTERVAL If f is integrable on the closed interval [a,b], then the average value of f on the interval is")
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FINDING THE AVERAGE VALUE OF A FUNCTION Find the average value of f(x) = 3x 2 – 2x on the interval [1,4]
![FINDING THE AVERAGE VALUE OF A FUNCTION Find the average value of f(x) = 3x 2 – 2x on the interval [1,4]](http://images.slideplayer.com./33/8262703/slides/slide_8.jpg "FINDING THE AVERAGE VALUE OF A FUNCTION Find the average value of f(x) = 3x 2 – 2x on the interval [1,4]")
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THE SPEED OF SOUND At different altitudes in Earth’s atmosphere, sound travels at different speeds. The speed of sound s(x) (in meters per second) can be modeled by where x is the altitude in kilometers. What is the average speed of sound over the interval [0,80]?
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THE DEFINITE INTEGRAL AS A FUNCTION Evaluate the function at x = 0, π/6, π/4, π/3, and π/2.
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THE SECOND FUNDAMENTAL THEOREM OF CALCULUS If f is continuous on an open interval I containing a, then, for every x in the interval,
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USING THE SECOND FUNDAMENTAL THEOREM OF CALCULUS
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THE NET CHANGE THEOREM The definite integral of the rate of change of a quantity F’(x) gives the total change, or net change, in that quantity on the interval [a,b]
![THE NET CHANGE THEOREM The definite integral of the rate of change of a quantity F’(x) gives the total change, or net change, in that quantity on the interval [a,b]](http://images.slideplayer.com./33/8262703/slides/slide_14.jpg "THE NET CHANGE THEOREM The definite integral of the rate of change of a quantity F’(x) gives the total change, or net change, in that quantity on the interval [a,b]")
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USING THE NET CHANGE THEOREM A chemical flows into a storage tank at a rate of 180+3t liters per minute, where 0 < t < 60. Find the amount of the chemical that flows into the tank during the first 20 minutes.
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DISPLACEMENT VS. DISTANCE Displacement on [a,b] = Total distance traveled on [a,b] =
![DISPLACEMENT VS. DISTANCE Displacement on [a,b] = Total distance traveled on [a,b] =](http://images.slideplayer.com./33/8262703/slides/slide_16.jpg "DISPLACEMENT VS. DISTANCE Displacement on [a,b] = Total distance traveled on [a,b] =")
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SOLVING A PARTICLE MOTION PROBLEM A particle is moving along a line so that its velocity is v(t) = t 3 – 10t 2 + 29t – 20 feet per second at time t. a.What is the displacement of the particle on the time interval 1 < t < 5? b.What is the total distance traveled by the particle on the time interval 1 < t < 5?
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HOMEWORK Page 306 #2, 3, 7-10, 13, 29, 31, 33, 37, 39, 47, 53, 107 Page 307 #62, 67-87 odd, 91, 129- 134 all
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