Thanks The area A of the region S that lies under the graph of the continuous fu

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The area A of the region S that lies under the graph of the continuous function is the limit of the sum of the areas of approximating rectangles. Use this definition to find an expression for the area under the graph of fas a limit. Do not evaluate the limit. Determine a region whose area is equal to the given limit. Do not evaluate the limit. Suppose the odometer on our car is broken and we want to estimate the distance driven over a 30 second time interval. We take the speedometer readings every five seconds and record them in the following table: In order to have the time and the velocity in consistent units, let's convert the velocity readings to feet per second (1 mi/h = 5280/3600 ft/s) During the first five seconds the velocity doesn't change very much, so we can estimate the distance traveled during that time by assuming that the velocity is constant. If we take the velocity during that time interval to be the initial velocity (23 ft/s), then we obtain the approximate distance traveled during the first five seconds: Similarly, during the second time interval the velocity is approximately constant and we take it to be the velocity when t - 5s. So our estimate for the distance traveled from t = 5s to t = 10s is If we add similar estimates for the other time intervals, we obtain an estimate for the total distance traveled We could just as well have used the velocity at the end of each time period instead of the velocity at the beginning as our assumed constant velocity. Then our estimate becomes If we had wanted a more accurate estimate, we could have taken velocity readings every two seconds, or even every second.

Explanation / Answer

The formula is (b-a)/n sum f(a+k(b-a)/n) -> int(a<=x<=b) f(x)dx for n->infinity

So :

First is last answer.

Second is last answer.

Last is : 29,43,115,145 and 1095/1080

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