Problem Set 9—Integration as Summation

Purpose

This lesson reinforces your understanding of the theoretical foundations of the definite integral and antiderivative as (infinite) sums.

Background

This exercise is mainly based on material in sections 5.1 through 5.3 of our textbook (“Area and Estimating with Finite Sums,” “Sigma Notation and Limits of Finite Sums,” and “The Definite Integral”). We covered (or will cover) this material in classes between approximately November 6 and November 13.

This exercise also draws to a small extent on knowledge of general antiderivatives, covered in class between November 6 and November 10, and discussed in section 4.7 (“Antiderivatives”) of our textbook.

Activity

Solve each of the following problems:

Problem 1

Find the most general antiderivative of the function f(x) = 5x³ + cosx.

Problem 2

A variation on section 5.1, exercise 2a: estimate the area under the curve f(x) = x³ between x = 0 and x = 1 using manual calculation of a lower sum with 2 rectangles of equal width. Then use R to make an estimate using a lower sum with 2000 rectangles of equal width.

Problem 3

Section 5.2, exercise 32c (evaluate the sum from k = 1 to k = n of k/(n²); express your answer in terms of n).

Problem 4

Section 5.2, exercise 42 (limit of Riemann sum of f(x) = 3x²; see textbook for details).

Problem 5

Section 5.3, exercise 10f (evaluate a definite integral of h(x) - f(x), given information about the integrals of h and f; see textbook for details).

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as that will speed the process along.

Sign up for a meeting via Google calendar. If you worked in a group on this exercise, the whole group should schedule a single meeting with me. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.