Problem Set 10 — Integrals

Purpose

This exercise mainly concentrates on your understanding of the foundations of the definite integral. In addition, it also touches briefly on L'Hôpital’s Rule. It therefore contributes to the following learning outcomes:

• Outcome 1. Compute limits of algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, and piecewise defined functions
• Outcome 2. Compute derivatives of algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, and piecewise defined functions
• Outcome 3. Compute definite integrals of algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, and piecewise defined functions
• Outcome 9. Solve problems in a range of mathematical applications using the integral
• Outcome 10. Apply the Fundamental Theorem of Calculus.

Background

We discussed the foundations of the definite integral, including its connections to the Fundamental Theorem of Calculus, in classes between November 15 and November 29. Our textbook talks about this material in sections 5.1 through 5.3. This material also relies on limits at infinity, which we talked about on November 12, and which is in section 4.6 of the textbook.

We talked about L'Hôpital’s Rule on November 12, and the textbook presents it in section 4.8.

Activity

Solve each of the following problems.

Problem 1

(Exercise 370 in section 4.8 of our textbook.) Evaluate the limit

Problem 2

(Based on problem 10 in section 5.1 of our textbook.) Use algebraic rules and closed form formulas for summations to evaluate the sum

Also write the sum as an explicit sum of numbers, although you don't have to evaluate this explicit sum. You may use a calculator for the purely numeric parts of this problem, but not to simplify the sum.

Problem 3

Consider the graph of \(y = \frac{1}{2} x\), and the area between that graph and the \(x\) axis between \(x = 1\) and \(x = 2\).

Part A

Express this area as a definite integral.

Part B

Estimate the value of the integral from Part A using a Riemann sum with 100 intervals. Use a right-endpoint approximation. (I recommend that you use closed form formulas and algebraic rules for summations to simplify the Riemann sum rather than just trying to evaluate it by brute force, although you can use a calculator for the final numeric calculations.)

Part C

Find the exact value of the integral from Part A by taking the limit of the Riemann sum from Part B.

Part D

Further confirm the exact value of the integral from Part A by evaluating it using the Evaluation Theorem (i.e., part 2 of the Fundamental Theorem of Calculus).

Part E

Finally, verify the area found via integration by using area formulas from geometry. (Consider plotting \(y = \frac{1}{2} x\) over the interval \(1 \le x \le 2\) if you want ideas about what formulas to use.)

Follow-Up

I will grade this exercise during an individual meeting with you. That meeting should happen on or before the “Grade By” date above. During the meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Sign up for the meeting via Google calendar. Please have a written solution to the exercise ready to share with me during your meeting, as that will speed the process along.