Comparing Left, Right, and Midpoint Riemann Sums
Investigate and approximate the area under the function from x = 0 to x = 2 Answer the questions below it to test your understanding.
Test your understanding: 1.) The Left Hand Sum will always give an underestimate of the actual area under f(x). How can you tell? What can you say about the Right Hand Sum? 2.) Notice on the interval , so f(x) is decreasing . Based on your answer to (1) above, explain how Left and Right Sums can give a good bound for the actual area under a decreasing function. What changes if the function was increasing? 3.) The Actual Area under f(x) from x=0 to x=2 is exactly, or 7.3333(rounded). Experiment with the number of rectangles. a.) How many rectangles is needed for the Left Sum to be within 0.1 of the actual area (7.3333)? What about for the Right Sum? and the Midpoint Sum? b.) Which type of Sum would you rather program in a computer to calculate the estimated area under a function? Why?