Math 1151, Autumn 2024

Written Homework 5

Assignment Goals: Using rectangles to approximate the area under a curve is the heart of the Riemann integral. This assignment will help you to practice:

❼ Using the notation and language of Riemann sums,

❼ Approximating the area under a curve using a large number of rectangles,

❼ Creating a formula for an approximation with a variable number of rectangles, and

❼ Using a limit to find the exact area under the curve.

Introduction:

Our goal is to understand the Riemann sum method for finding area under a curve. This homework will walk you through this process. In Problem 1, we will find a formula for a Riemann sum using a variable n number of rectangles. Then in Problem 2, we use a limit to find the exact area under the curve.

Before beginning this homework, you should practice finding a Riemann sum using only a few rectangles. Examples of this can be found in our Ximera textbook, Ximera homework, and recitation worksheets. If you have any questions about the terminology or notation we use, make sure to ask your instructors!

Remember to show your work 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.

Problem 1: Finding an Approximation with a Variable Number of Rectangles (40 points)

For this homework, consider the function f (x) = 2/1x2 − 2x + 6. A partial graph is given above.

In this problem, we will find the Riemann sum of f (x) over the interval [0, 6] using n rectangles and right endpoints.

a) (2 points) Find a formula for the width, ∆x, of each rectangle used in this approximation.

(Your answer should contain the variable n.)

∆x =                                                                              

b) (3 points) In your own words, explain what role grid points play in the Riemann Sum process. Why do we need to find grid points?

c) (3 points) Find the first three grid points, x0, x1, and x2. (We assume n ≥ 2, but leave n as a variable.)

x0 =                                          x1 =                                       x2 =                          

d) (4 points) Find a formula for the grid point xk , where k can be any integer satisfying 0 ≤ k ≤ n.

(Your answer should contain the variables k and n.)

xk =                                                                                

e) (3 points) In your own words, explain what role sample points play in the Riemann Sum process. Why do we need to find sample points?

f) (3 points) Find the first three sample points, x*1, x*2, and x*3.

(We assume n ≥ 3, but leave n as a variable. Remember that we are using right endpoints.)

x*1 =                                      x*2 =                                     x*3 =                                  

g) (4 points) Find a formula for the sample point xk ∗ , where k can be any integer satisfying 1 ≤ k ≤ n.

(Your answer should contain the variables k and n. Remember that we are using right endpoints.)

x*k =                                                                                

h) (6 points) Find a formula for the height, width, and area of the k’th rectangle, where k is any integer satisfying 1 ≤ k ≤ n. (Your answers should contain the variables k and n. Your answer should not contain the letter f . If you use the function f (x), you should evaluate it.)

Width of k’th rectangle:                                                               

Height of k’th rectangle:                                                              

Area of k’th rectangle:                                                                 

i) (4 points) Write the Riemann sum for this area approximation in Sigma notation.

(Do not evaluate the sum here. Your answer should contains the variable n and the index variable k.)

Sum in Sigma Notation:                                                               

j) (8 points) Evaluate the sum from part i) above.

(Your answer should contain the variable n. Use the summation formulas from this section in the textbook where appropriate.)

Evaluated Sum:                                                                     

Problem 2: Using the Formula Found in Problem 1 (20 points)

In Problem 1, we found a formula for the Riemann sum using n rectangles. Since we left n as a variable, we can now plug in any integer we like for n. In this way, we quickly find the Riemann sum using large numbers of rectangles.

a) (6 points) Use the formula found in Problem Problem 1(j) to find the following Riemann sums.

Answers should not be simplified.

Riemann sum for 4 rectangles:                                                 

Riemann sum for 400 rectangles:                                              

Riemann sum for 40000 rectangles:                                           

Next we ask you to take a limit of the Riemann sum as n goes to ∞. Take a moment to appreciate the complexity of what we want to do. Every time we add another rectangle, the width and height of every rectangle changes. All of these details factor into the formula you came up with.

b) (10 points) Take the limit as n → ∞ of the Riemann sum found in Problem 1(j). (Your final answer should have no variables left in it. Don’t forget to check the form. and show your work. You may use l’Hopital’s Rule.)

Limit Value:                                                                

c) (4 points) Explain what the value of this limit from Part (b) represents in the context of this problem. Be as specific as you can (i.e. do not say ”the function”, instead give the name of the function.)