1. Math 180 5.1 – Integration 1

2. We know how to calculate areas of some shapes. 2

3. Can we find a more general way to figure out areas? Let’s try looking at areas under curves… 3

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12. 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

13. 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

14. 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

15. 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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23. 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

24. 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

25. 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

26. 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

27. We could also use the midpoint of each subinterval to get the heights of the rectangles: 27

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38. 38 Distance Traveled

39. 39 Distance Traveled

40. 40 Distance Traveled

41. 41 Distance Traveled (40 mph)(2 hours) = 80 miles

42. 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

43. Ex 1. Estimate the distance traveled given the following sample velocities. 43 Time (s)051015202530 Velocity (ft/s)25313543474641

44. We can visualize our calculations on a coordinate system: 44

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46. 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