04 - Approx - Oil is leaking out of a container at a...

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Approximations So far we can find areas of squares, parallelograms, triangles and polynomials (which we can break into smaller triangles). Now we want to expand the number of curves for which we can find area.
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f ( x i ) Δ x i ` n is a Riemann sum Ex. Approximate the area above the x-axis and below y x between x = 1 and x = 4. Let n = 3
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Note: The midpoint average ≠ the average of the left and right approximations Ex. Ex. Use n = 6 to approximate the area under y x .
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The exact area = lim n ± f x i i 1 n ± V x Ex. Suppose we have the velocity of a car from t = 0 to t = 5 sec. Find the distance that the car traveled in 5 seconds. sec t 0 1 2 3 4 5 ft/sec v 20 33 39 43 47 50 Distance = (velocity)(time) (ft/sec)(sec) = ft
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Ex. Let Smax denote the largest Rieman sum that can be formed for the function f on the given interval using the following table of values: x 0 .3 .6 .9 1.2 1.5 1.8 f(x) 1.5 2 2.5 1 0 -.5 .5 Smax =
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Do:
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Unformatted text preview: Oil is leaking out of a container at a decreasing rate. Time (hrs.) 2 4 6 8 Rate (liters/hr.) 35 26 23 19 18 We want to estimate the total amount of oil that has leaked out of the container during the time shown. 1. What is the maximum amount of oil that could have leaked out during the first two hours? The minimum? 2. What is the maximum amount of oil that could have leaked out during the second two hour period(i.e. between t = 2 and t = 4)? The minimum? 3. During the entire eight-hour interval from t = 0 to t = 8, what is the maximum amount of oil that could have leaked out? the minimum? Explain where you used the assumption that the rate at which the oil was leaking was decreasing. 4. If you had to guess how much oil actually leaked out, what would you guess?...
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04 - Approx - Oil is leaking out of a container at a...

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