Week In Review #8 Covers sections: 5.1, 5.2, 5.3 and 5.4. Things you must know

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1 Week In Review #8 Covers sections: 5.1, 5.2, 5.3 and 5. Things you must know Know how to get an accumulated change by finding an upper or a lower estimate value Know how to approximate a definite integral using Riemann Sums: (LHS, RHS, n, a, b, and x..) Know how to find a definite integral using a graphing calculator Know how to find an area under a function or between two functions Know how to interpret an integral Know how to find the units of an integral Review Problems: 1. Roger runs a marathon. His friend Jeff rides behind him on a bicycle and clocks his speed. Roger starts out strong, but after an hour and a half he is so exhausted that he has to stop. Jeff's data is as follow: Time since start (min) Speed (mph) (a) Find the upper and lower estimates for the total distance Roger ran during the first half hour. (b) Estimate the total distance Roger ran in the marathon. (c) Sketch a graph representing the upper estimate.

2 2. The rate of sales (in CDs per week) of a new CD is shown in the following table. t (weeks) Rate of sales (CDs per week) (a) Find an upper estimate of the total number of CDs sold throughout the -week period. (b) Find a lower estimate of the total number of CDs sold throughout the -week period. (c) Find a Right-hand sum to estimate of the total number of CDs sold throughout the - week period. (d) Find a left-hand sum to estimate of the total number of CDs sold throughout the - week period. (e) Estimate the total number of CDs sold throughout the -week period. 3. The following figure shows the graph of the velocity, v (miles per hour), of a car traveling in the same direction along a straight road. v (mph) t (hr) (a) Estimate the total distance the car traveled between t = 0 and t =.5. (c) Find the area between v and the t-axis from t = 0 to t =.5. Interpret your answer.

3 . Using the figure below, draw rectangles representing each of the following Riemann Sums for the function f on the interval 0 x 10 and calculate the value of each sum. (a) Left-hand sum with x = 5 (b) Right-hand sum with x = (c) Left-hand sum with x = 2 (d) Right-hand sum with x = Estimate 10 f ( xdx ) using the above figure with n = 5. 0

4 6. Use the following table to estimate Htdt (). What are n and t? 3 t H(t) Estimate the value of the definite integral 3 x dx using a left-hand sum with n =. Is this a 1 lower or upper estimate? 8. (a) Use the following graph to find f ( x) dx. (b) Find the area between f(x) and the x-axis from x = -3 and x = t e 9. Evaluate the definite integral dt using a calculator. 2 2 t t

5 2 10. Find the area under the graph of f ( x) = x x from x = -1 to x =. 11. Oil is leaking from a tanker at a rate of 10(.7) t gallons per minute, with t in minutes. (a) Find the total quantity of oil that leaks out in the first three minutes. (b) Does more oil leak out during the first minute or the third minute? Justify the answer graphically. 2. Given two functions f ( x) = x and gx ( ) = 3sin( x), (a) sketch the functions and shade the area bounded by the functions from x = 0 to x = π, (b) set up the integral(s) that represent the area in (a), (c) and compute the shaded area to 2 decimal places.

6 13. If f(t) is measured in meters/second 2 and t is measured in seconds, what are the units of f ( t) dt? b a 1. Blood concentration curves of two drugs are shown below. Definition: Bioavailability Concentration Time Concentration of drug in blood stream Drug A Drug B t (hr) (a) Which drug has the largest concentration? (b) Which drug has bigger Bioavailability at the end of the first hour? (c) Which drug has the bigger Bioavailability over all? 15. Use the graph below to arrange the definite integrals from smallest to largest. f(x) a b c b c c c b A= f( x) dx B= f( x) dx C = f( x) dx D= f( x) dx E = f( x) dx a b a a b

7 Answers: 1. (a) The upper estimate is 5.75 miles and the lower estimate is 5.25 miles during the first half hour. (b) The total distance is around 13 miles. 2. (a) 26,250 CDs (b) 15,500 CDs (c) 2,625 CDs (d) 17,5 CDs (e),875 CDs 3. (a) The total distance is 97.5 miles. (b) The area is 97.5 which means the total distance the car traveled between t = 0 and t =.5 equals the area between the curve of v and t-axis from 0 to.5.. (a) 210 (b) 130 (c) 192 (d) n = 5, t = 0.2, and the integral equals LHS = 2.61.The left-hand-sum is a lower estimate for 3 x dx (a) 5.5 (b) (a) 18.2 gallons (b) More oil leaks out during the first minute.. (b) π 2 2 (3sin x x ) dx+ ( x 3sin x) dx (c) meters/second 1. (a) Drug A (b) Drug A (c) Drug B 15. B < E < C < A < D If you find any mistakes, please let me know. Thanks! li-chen2@neo.tamu.edu