Problem 1:

1 For the following problems. consider the function f(x} = 5x2 -- 2X l- 4 over the interval [0. 5]. The graph is given below. Our goal is to find the area under this curve over the interval [0.6]. We will do this in a Few stages. In Problem 1. we will make an estimate using a small number of rectangles. in Problem 2. we will get a better estimate by using a large number of rectangles. In Problem 3. we will find a formula for an approximation using a variable number of rectangles. And in Problem 4. we use a limit to find the exact area under the curve. Remember to show your lwrork on each part of this homework. Even if the calculation is simple. your work must show how you got your answer. including any formulas you used. 12 H 7 Problem 1: Approximating Area with Three Rectangles {15 points] In this problem. we want to approximate the area under the graph of y = {(x}. above the x-axis. over the interval [0. 6] by using in = 3 rectangles and right endpoints. a} (2 points} What is the width. x. of each rectangle used in this approximation? Ax .. b} (2 points} List the grid points. x0. x1. x2. x3. used in this approximation. Grid points: c} (3 points} List the sample points. x-f. x.X'. used in this approximation. (Remember that we are using right endpoints} Sample points: d) (3 points} Illustrate this area approximation by sketching the corresponding rectangles in the graph above. e} (5 points} Calculate this area approximation. Area Approximation: Math 1151, Spring 2023 Written Homework 5 Page 4 of 7 d) (2 points) Find a formula for the height of the k'th rectangle, f(x*). (The only variable appearing in your expression should be k.) f ( x* ) = e) (3 points) Write the Riemann sum for this area approximation in Sigma notation. (Do not evaluate the sum here. The only variable appearing in your expression should be the index variable k.) Sum in Sigma Notation: f) (6 points) Evaluate the sum from part (e) above. ( Your answer should have no variables remaining. Use the summation formulas from this section in the textbook where appropriate.) Evaluated Sum:Problem 2: Finding a Better Approximation with Thirtyr Rectangles (13 points] Finding a Riemann Sum using only three rectangles does not require much work. but it also does not give us a great approx- imation. To nd a better approximation we need to use more rectangles. The trade-off is a more complicated calculation. Instead of finding every grid and sample point we will find a general formula the points. This problem will walk you through this process. In this problem. we want to approximate the area under the graph of y = f(x}. above the xaxis. over the interval [0. 6} by using n = 30 rectangles and right endpoints. a} (2 points} Find a formula for the width. x. of each rectangle used in this approximation. Ax: b) (3 points) Find a formula for the grid point xk. (The only variable appearing in your expression should be (1.) X :- || c} (2 points} Find a formula for the sample point x;. (The only variable appearing in your expression should be it. Remember that we are using right endpoints.) ,5