Applications to Physics and Engineering

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 6.4 – Work
• Section 7.7 – Approximate Integration
• Section 8.3 – Applications to Physics and Engineering

Expected Educational Results

• Objective 14–1: Given the velocity of an object along a straight line, I can set up the integral to compute the displacement and total distance traveled of the object.
• Objective 14–2: Given the acceleration of an object along a straight line, I can set up the integral to compute the velocity, speed, displacement, and total distance traveled of the object.
• Objective 14–3: I can estimate a definite integral using a Riemann Sum.
• Objective 14–4: I can use the shape of a function to determine if a Riemann Sum is an over- or under-estimate of a definite integral.
• Objective 14–5: I can set up the integral to compute the moments and center of mass of a lamina.
• Objective 14–6: I can set up and evaluate the integral to compute the work on an object.
• Objective 14–7: I can set up and evaluate the integral using Hooke's Law to compute the work on a spring.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Approximate Integration

NOTE: On all assessments, you will be given access to the DESMOS graphs included in this document.

Riemann Sums

DESMOS Graphs

Left, Right, and Midpoint Sums

Trapezoid Sum

Simpson Sum

Question 10

Use the above DESMOS links to find the following:

1. Estimate (to 2 decimal places} ${\displaystyle {\int }_1^2{e}^{-x^2}dx}$$\displaystyle \int_1^2{e^{-x^2}\,dx}$ using $L_4$$L_4$.

   1. Is $L_4$$L_4$ an overestimate? an underestimate?
   2. Use the shape of the curve to explain.

2. Estimate (to 2 decimal places} ${\displaystyle {\int }_0^1{x}^2\sin (x^2)dx}$$\displaystyle \int_0^1{x^2\sin{(x^2)}\,dx}$ using $R_4$$R_4$.

   1. Is $R_4$$R_4$ an overestimate? an underestimate?
   2. Use the shape of the curve to explain.

3. Estimate (to 2 decimal places} ${\displaystyle {\int }_0^1\frac{1}{x^3+1}dx}$$\displaystyle \int_0^1{\frac{1}{x^3+1}\,dx}$ using $M_4$$M_4$.

   1. Is $M_4$$M_4$ an overestimate? an underestimate?
   2. Use the shape of the curve to explain.

4. Estimate (to 2 decimal places} ${\displaystyle {\int }_5^8{\cos }^3\left(\frac{x}{3}\right)dx}$$\displaystyle \int_5^8{\cos^3{\left(\frac{x}{3}\right)}\,dx}$ using $T_4$$T_4$.

   1. Is $T_4$$T_4$ an overestimate? an underestimate?
   2. Use the shape of the curve to explain.

5. Estimate (to 2 decimal places} ${\displaystyle {\int }_0^5{e}^{-x/10}{\cos }^3\left(\frac{x}{2}\right)dx}$$\displaystyle \int_0^{5}{e^{-x/10}\cos^3{\left(\frac{x}{2}\right)}\,dx}$ using $S_4$$S_4$.

   1. Is $S_4$$S_4$ an overestimate? an underestimate?
   2. Use the shape of the curve to explain.

Homework

At this time, you should be able to complete the following assignments:

• Section 7.7: # 1, 3, 7, 9, 11, 13, 15.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Last Modified: Thursday, 15 October 2020 6:42 EDT