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

Consider the integral ${\displaystyle {\int }_a^{b}f(x)dx}$$\displaystyle \int_a^b{f(x)\,dx}$.

Left-hand Sum

1. Let $n$$n$ be the number of partitions of the area under the $f(x)$$f(x)$ curve;
2. Let $\mathrm{\Delta }x=\frac{b-a}{n}$$\Delta x =\frac{b-a}{n}$ be the width of each partition;
3. Let $x_i$$x_i$ be the sample points (i.e., left-hand of partition).

Definition: Left-hand sum

The left-hand sum is defined as ${\displaystyle L_n=\sum _{i=0}^{n-1}f(x_i)\mathrm{\Delta }x}$$\displaystyle L_n=\sum_{i=0}^{n-1}{f(x_i)\Delta x}$

Investigation 05

To visualize $L_n$$L_n$ use the embedded DEMSOS graph below.

Use $c=0$$c=0$.

1. Where does each rectangle touch the curve?
2. Increase the value for $n$$n$ by moving the slider for $n$$n$. What do you notice? Explain.

Right-hand Sum

1. Let $n$$n$ be the number of partitions of the area under the $f(x)$$f(x)$ curve;
2. Let $\mathrm{\Delta }x=\frac{b-a}{n}$$\Delta x =\frac{b-a}{n}$ be the width of each partition;
3. Let $x_i$$x_i$ be the sample points (i.e., right-hand of partition).

Definition: Right-hand sum

The right-hand sum is defined as ${\displaystyle R_n=\sum _{i=1}^{n}f(x_i)\mathrm{\Delta }x}$$\displaystyle R_n=\sum_{i=1}^{n}{f(x_i)\Delta x}$

Investigation 06

To visualize $R_n$$R_n$ use the embedded DEMSOS graph below.

Use $c=1$$c=1$.

1. Where does each rectangle touch the curve?
2. Increase the value for $n$$n$ by moving the slider for $n$$n$. What do you notice? Explain.

Midpoint Sum

1. Let $n$$n$ be the number of partitions of the area under the $f(x)$$f(x)$ curve;
2. Let $\mathrm{\Delta }x=\frac{b-a}{n}$$\Delta x =\frac{b-a}{n}$ be the width of each partition;
3. Let ${\overline{x}}_i=\frac{x_{i-1}+x_i}{2}$$\bar{x}_i=\frac{x_{i-1}+x_i}{2}$ be the sample points (i.e., midpoint of partition).

Definition: Midpoint sum

The midpoint sum is defined as ${\displaystyle M_n=\sum _{i=1}^{n}f({\overline{x}}_i)\mathrm{\Delta }x}$$\displaystyle M_n=\sum_{i=1}^{n}{f(\bar{x}_i)\Delta x}$

Investigation 07

To visualize $M_n$$M_n$ use the embedded DEMSOS graph below.

Use $c=0.5$$c=0.5$.

1. Where does each rectangle touch the curve?
2. Increase the value for $n$$n$ by moving the slider for $n$$n$. What do you notice? Explain.

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