AP Calculus
Riemann Sums and Trapezoidal Rule
Name: _____________________
1. By reading values from the given graph of f, use five rectangles to find: a. a lower estimate for the area under the given graph b. an upper estimate for the area under the given graph of f from x = 0 to x = 10. Sketch the rectangles of f from x = 0 to x = 10. Sketch the rectangles that y you use.










 x that you use.










 y x
             


           graph of f from x = 0 to x = 10. Sketch the rectangles
 that you use.

 y
















             

           d. an estimate using trapezoids for the area under the
 trapezoids that you use.
 y




 
 


 
 




 

             


        x
                



        x
  
________________________________________________________________________________________________
 f ( x )





1
1
2. Graph the function
 x
2 a. Right endpoints
 three approximating rectangles and taking the sample points to be:

 b. Left endpoints c. Midpoints





 d. Trapezoids
 
 
 
 
 
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3. Coal gas is produced at a gasworks. Pollutants are removed by screens which become less efficient as time goes on. Measurements are made every two months showing the rate at which pollutants escape..
Time (months) 0
Rate (tons/month) 5
2
8
4 6
13 20
Find the amount of pollutants that escape using a: a. lower estimate b. upper estimate
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4. The following table gives the emissions, E, of nitrogen oxides in millions of metric tons per year in the US. Let t be the number of years since 1940 and E
 f ( t ) .
Year
E
1940
6.9
1950
9.4
1960
13.0
1970
18.5
1980
20.9
1990
19.6
Estimate the amount of nitrogen oxides emitted from 1940 to 1990 using five subintervals and: a. a left Riemann sum b. a right Riemann sum. c. a trapezoidal approximation
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5. Oil is leaking out of a tanker damaged at sea. The damage to the tanker is worsening as evidenced by the increased leakage each hour, recorded in the table below.
Time (hours) 0
Leakage (gal/hr) 50
1
70
3 6 8
136 369 720 a. Give an upper estimate for the total quantity of oil that has escaped after 8 hours. b. Give a lower estimate for the total quantity of oil that has escaped after 8 hours.

6. The graph of the velocity v(t), in ft/sec, of a bicycle racing on a straight road, for a. Approximate the value of the distance the bicycle traveled using a
 y
right Riemann sum with the five subintervals indicated by the data in the graph. Show your setup.


0
 t

60 , is shown below. b. Is this numerical approximation less than the actual distance the bicycle traveled? Explain your reasoning.





 c. Approximate the value of the distance the bicycle traveled using a left Riemann sum with the five subintervals indicated by the data in the graph. Show your setup.



          
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 x
7. Ben Travlen rides a unicycle back and forth along a straight east-west track. The twice differentiable function models Ben’s position on the track, measured in meters from the western end of the track, at time
B t , measured in seconds from the start of the ride. The table gives values for Ben’s velocity, v ( t ) , measured in meters per second, at selected times t .
(A) Use the data in the table to approximate Ben’s acceleration at time t

5 seconds. Indicate units of measure.
(B) Approximate the distance Ben traveled for the first 60 seconds using a left Riemann sum with the subintervals indicated by the data in the table.
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8. The table shows selected values for a twice differentiable function f. x f(x)
-3
5
-2
3
-1
2
0
-1
1
1
2
-4
3
-7
Estimate the area under the graph of f using a midpoint Riemann sum with 3 subintervals of equal length.
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9. The function R that approximates the rate that water is being pumped into a reservoir, in gallons per minute, is continuous on the closed interval [ 0 , 6 ] and has the values given in the table. The trapezoidal approximation for the amount of water pumped for the first 6 hours found with 3 subintervals of equal length is 52. What is the value of k ? t (minutes) 0
Rate 4
(gal/min)
2
K
4
8
6
12
(A) 2 (B) 6 (C) 7 (D) 10 (E) 14
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10. The graph shows the rate of bamboo growth bamboo is 60 cm tall at time t

10 g ( t ) in centimeters (cm) per day over a 20 day period. If the
days, approximately how tall is it at t

20 days?
(A) 25 cm
(B) 64 cm
(C) 70 cm
(D) 82 cm
(E) 100 cm