Calculus AB Assignment Finding a Better Approximation of Area Under a Curve You'll be working with the function y =x} = {x 1.5}? + 2. and the interval you'll always be concerned about is [-2.6]. Here's a graph: You'll need to have the graph {above} on paper, so you'll need to print it or draw it on paper. You'll need at least three copies. I'll call them your "paper graphs." The actual area under the curve on the interval [2.6] is about 11536. Approximating the area underthe curve on the interval [2,6] by using a Riemann sum with 4 equal subdivisions and left-hand endpoints gives you R = Elm-2} +f{l]) +2) +4J) = 14.976. Draw the rectangles used for this Riemann sum on one of your paper graphs. Do you see where the error for this approximation is? Approximating the area underthe curve on the interval [2.6] by using a Riemann sum with 4 equal subdivisions and right-hand endpoints gives you R = 2MB) +2) +f[4) +f(6:i) = 21.314. Draw the rectangles used for this Riemann sum on one of your paper graphs. Do you see where the error for this approximation is? Now, you'll try to improve these approximations. (Remember, the interval you're always working with is [4,6].) 1. Re-oompute the Riemann sum, but this time increase the number of subdivisions. Take 11 to be at least 8 {with equal subdivisions} and compute a Riemann sum using either left-hand endpoints or right-hand endpoints. Show all your computation work and compare the approximation to the Riemann sum using only 4 subdivisions and the same endpoint scheme. (Note: You don't need to show function evaluations, so it's okay to write, for example,l2} +j{13) +f(14} = [unt-fr], and let your calculator compute the answer, as long as its correct.) 2. Use midpoints. Take it to be 4 with equal subdivisions and compute a Riemann sum using midpoints. Show all your work and compare the approximation to the Riemann sums with left-hand and right-hand endpoints. 3. Now, still take a to be 4, but don't be restricted by having to take equal subdivisions. You should use one of your paper graphs to try to draw the 4 best rectangles to approximate the area under the curve. Then add up the areas of the rectangles and what do you get? Show all your work, indicating where you placed your subintervals and how you arrived at the heights of the rectangles. Include your graph. How does the approximation compare with the others you've made? 4. Finally. devise a way to improve the approximation of area under this curve, still taking only 4 subdivisions. (Hint: You don't need to use rectangles if you don't want to.) Explain your method and calculations. Is your approximation any better than the previous ones? Do you think your method will work well in general. for example, fora general curve? There is no one correct answer here. The important thing is that you come up with an alternative idea and that you explain how it works and why it works better than a Riemann sum