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Math 180 5.1 – Integration 1
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We know how to calculate areas of some shapes. 2
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Can we find a more general way to figure out areas? Let’s try looking at areas under curves… 3
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When the heights of the rectangles are determined by the _________ function value on each subinterval, our area estimation is an __________, and it always ______________ the true area under the curve. 12
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When the heights of the rectangles are determined by the _________ function value on each subinterval, our area estimation is an __________, and it always ______________ the true area under the curve. 13 maximum
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When the heights of the rectangles are determined by the _________ function value on each subinterval, our area estimation is an __________, and it always ______________ the true area under the curve. 14 maximum upper sum
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When the heights of the rectangles are determined by the _________ function value on each subinterval, our area estimation is an __________, and it always ______________ the true area under the curve. 15 maximum upper sumoverestimates
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When the heights of the rectangles are determined by the _________ function value on each subinterval, our area estimation is a __________, and it always ______________ the true area under the curve. 23
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When the heights of the rectangles are determined by the _________ function value on each subinterval, our area estimation is a __________, and it always ______________ the true area under the curve. 24 minimum
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When the heights of the rectangles are determined by the _________ function value on each subinterval, our area estimation is a __________, and it always ______________ the true area under the curve. 25 minimum lower sum
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When the heights of the rectangles are determined by the _________ function value on each subinterval, our area estimation is a __________, and it always ______________ the true area under the curve. 26 minimum lower sumunderestimates
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We could also use the midpoint of each subinterval to get the heights of the rectangles: 27
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38 Distance Traveled
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39 Distance Traveled
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40 Distance Traveled
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41 Distance Traveled (40 mph)(2 hours) = 80 miles
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If velocity is not constant, then there’s no easy formula. But we can take sample readings of velocity at a bunch of time intervals, and make estimations of the distances traveled on each time interval based on our sample velocities. 42
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Ex 1. Estimate the distance traveled given the following sample velocities. 43 Time (s)051015202530 Velocity (ft/s)25313543474641
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We can visualize our calculations on a coordinate system: 44
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This gives us our first clue that the area under the curve of a function is somehow related to the antiderivative of the function… 46
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