In this worksheet you will practice computing and comparing estimates of the area bound by a...

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In this worksheet you will practice computing and comparing estimates of the area bound by a graph and an interval on the r-axis. As an example, we will consider the region bound by the graph y = 2 +9 over the interval [0, 1] on the r-axis. Let R denote this region and A denote its area. 1. Use a calculator to sketch the graph y = +9 over the interval [0, 1] twice. That is, sketch R twice. Use your first sketch to illustrate how R is approximated when we use a midpoint Riemann sum with five subintervals. Use the second sketch to illustrate how R is approximated when we use the trapezoid rule with five subintervals. Label cach graph with a detailed caption. Date: September 20, 2022. 2 1920, PRACTICE APPROXIMATING INTEGRALS AS SUMS 2. Compute the midpoint Riemann sum and trapezoid rule sum that you illustrated in exercise 1. Note that the value of each sum approximates A. Let A denote the value of the midpoint Riemann sum and Ar denote the value of the trapezoid rule sum. Round each value to 4 decimal places. 3. a) Use the error bounds from the text/notes to get an upper bound on the absolute value of the error between A and AM. Let BM denote this bound. b) Use the error bounds from the text/notes to get an upper bound on the absolute value of the error between A and AT. Let BT denote this bound. 1920, PRACTICE APPROXIMATING INTEGRALS AS SUMS 4. Compute the exact value of A. Round your answer to four decimal places. 3 5. a) Compute the absolute value of the error between A and AT. That is, compute |ET|= |AAT. Round your answer to three decimal places. Compare |ET| and BT. Which is greater? Explain. b) Compute the absolute value of the error between A and AM. That is, compute |EM| = |A-AM|. Round your answer to three decimal places. Compare EM and BM- Which is greater? Explain. 5. Use the error bounds from the text/notes to determine how many subintervals would be sufficient to guarantee that: a) ET < 0.01 b) EM < 0.01 In this worksheet you will practice computing and comparing estimates of the area bound by a graph and an interval on the r-axis. As an example, we will consider the region bound by the graph y = 2 +9 over the interval [0, 1] on the r-axis. Let R denote this region and A denote its area. 1. Use a calculator to sketch the graph y = +9 over the interval [0, 1] twice. That is, sketch R twice. Use your first sketch to illustrate how R is approximated when we use a midpoint Riemann sum with five subintervals. Use the second sketch to illustrate how R is approximated when we use the trapezoid rule with five subintervals. Label cach graph with a detailed caption. Date: September 20, 2022. 2 1920, PRACTICE APPROXIMATING INTEGRALS AS SUMS 2. Compute the midpoint Riemann sum and trapezoid rule sum that you illustrated in exercise 1. Note that the value of each sum approximates A. Let A denote the value of the midpoint Riemann sum and Ar denote the value of the trapezoid rule sum. Round each value to 4 decimal places. 3. a) Use the error bounds from the text/notes to get an upper bound on the absolute value of the error between A and AM. Let BM denote this bound. b) Use the error bounds from the text/notes to get an upper bound on the absolute value of the error between A and AT. Let BT denote this bound. 1920, PRACTICE APPROXIMATING INTEGRALS AS SUMS 4. Compute the exact value of A. Round your answer to four decimal places. 3 5. a) Compute the absolute value of the error between A and AT. That is, compute |ET|= |AAT. Round your answer to three decimal places. Compare |ET| and BT. Which is greater? Explain. b) Compute the absolute value of the error between A and AM. That is, compute |EM| = |A-AM|. Round your answer to three decimal places. Compare EM and BM- Which is greater? Explain. 5. Use the error bounds from the text/notes to determine how many subintervals would be sufficient to guarantee that: a) ET < 0.01 b) EM < 0.01

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