{"id":462,"date":"2025-02-13T19:45:12","date_gmt":"2025-02-13T19:45:12","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/physical-applications-of-integration-background-youll-need-3\/"},"modified":"2025-02-13T19:45:12","modified_gmt":"2025-02-13T19:45:12","slug":"physical-applications-of-integration-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/physical-applications-of-integration-background-youll-need-3\/","title":{"raw":"Physical Applications of Integration: Background You'll Need 3","rendered":"Physical Applications of Integration: Background You&#8217;ll Need 3"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Recognize the fundamental concepts of reflection, rotation, and translation symmetry<\/li>\n<\/ul>\n<\/section>\n<h2>Symmetry<\/h2>\n<p><strong>Symmetry<\/strong>, in its simplest form, refers to a sense of harmonious and aesthetically pleasing proportionality and balance. It's a principle that is deeply rooted in nature, mathematics, art, design, and architecture. The basic idea behind symmetry is that if you were to draw a line through an object, shape, or design, one side would be a mirror image of the other.<\/p>\n<center><img class=\"aligncenter wp-image-5043\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/53\/2025\/02\/13194508\/Asymmetric_PSF.svg_.png\" alt=\"Two drawings of a tree are show, one with a line creating a symmetrical image and another with a line created an asymmetrical iamge\" width=\"500\" height=\"310\"><\/center>\n<p>&nbsp;<\/p>\n<p>The line or plane that divides the shape, object or design into two equal and mirrored halves is called the <strong>axis of symmetry<\/strong>. This axis can be vertical, horizontal, or diagonal, depending on the orientation of the symmetrical shape.<\/p>\n<p>In visual arts, symmetry is used to create a sense of balance and harmony within a composition. Symmetrical compositions can be calming and stable, often conveying a sense of order and perfection.<\/p>\n<p>However, perfectly symmetrical compositions can also risk being seen as static or boring, which is why artists often balance symmetry with elements of asymmetry to create interest and dynamism. Breaking symmetry, while maintaining balance, can create visual interest and help guide the viewer's eye to a focal point. This is often used in visual arts and design to create tension, emphasize certain elements, or convey movement and change.<\/p>\n<p>There are three primary types of symmetry - <strong>reflection<\/strong>, <strong>rotation<\/strong>, and <strong>translation symmetry<\/strong> - which are often used in different combinations to create aesthetically pleasing and balanced designs.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>reflection symmetry<\/h3>\n<p><strong>Reflection symmetry<\/strong>, also known as mirror or bilateral symmetry, occurs when one half of an image, shape or design is the mirror image of the other half. That is, one side reflects the other.<\/p>\n<p>&nbsp;<\/p>\n<center><img class=\"size-medium wp-image-5047\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/53\/2025\/02\/13194509\/Symmetry-300x294-1.png\" alt=\"a set of four different shapes, three with reflection symmetry and one that is asymmetrical\" width=\"300\" height=\"294\"><\/center><center><strong><span style=\"font-size: 10pt;\">Figures with the axes of symmetry drawn in. The figure with no axes is asymmetric.<\/span><\/strong><\/center><\/div>\n<\/section>\n<p>Reflection symmetry is common in architecture and design, as it imparts a sense of balance and harmony. Artists may also use it to create emphasis on a particular aspect of their work. For instance, think of a butterfly. The right wing is a mirror image of the left wing, hence it has reflection symmetry. In architecture, many buildings and structures - from the Taj Mahal to modern skyscrapers - feature reflection symmetry.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>rotation symmetry<\/h3>\n<p><strong>Rotation symmetry<\/strong>, or radial symmetry, happens when a design or image can be rotated around a central point and still appear the same. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n[caption id=\"attachment_5049\" align=\"alignleft\" width=\"150\"]<img class=\"size-thumbnail wp-image-5049\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/53\/2025\/02\/13194510\/683px-Finland_road_sign_166_1995%E2%80%932020.svg_-150x150-1.png\" alt=\"A triangular sign with three degrees of rotation symmetry\" width=\"150\" height=\"150\"> An example of three degrees of rotation symmetry.[\/caption] [caption id=\"attachment_5050\" align=\"alignleft\" width=\"150\"]<img class=\"size-thumbnail wp-image-5050\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/53\/2025\/02\/13194510\/Cyclic_symmetry_4-150x150-1.png\" alt=\"A drawing with four degrees of rotational symmetry\" width=\"150\" height=\"150\"> An example of four degrees of rotation symmetry.[\/caption] [caption id=\"attachment_5051\" align=\"alignleft\" width=\"150\"]<img class=\"size-thumbnail wp-image-5051\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/53\/2025\/02\/13194511\/599px-15crossings-decorative-knot.svg_-150x150-1.png\" alt=\"A drawing with five degrees of rotation symmetry\" width=\"150\" height=\"150\"> An example of five degrees of rotation symmetry.[\/caption] [caption id=\"attachment_5052\" align=\"alignleft\" width=\"150\"]<img class=\"size-thumbnail wp-image-5052\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/53\/2025\/02\/13194511\/459px-Olavsrose.svg_-150x150-1.png\" alt=\"A drawing with six degrees of rotational symmetry\" width=\"150\" height=\"150\"> An example of six degrees of rotation symmetry.[\/caption]\n<\/section>\n<p>This type of symmetry is common in nature, such as in flowers and snowflakes, and is often used in design and art to create patterns that convey movement and dynamism.<\/p>\n<section class=\"textbox example\">\n<p>A designer is creating a circular tile pattern with sixfold rotation symmetry for a floor. If the radius of the circle is [latex]1.5[\/latex] meters, calculate the area of one segment (slice) of the pattern.<\/p>\n<p><br>\n[reveal-answer q=\"872000\"]Show Answer[\/reveal-answer]<br>\n[hidden-answer a=\"872000\"]The area of the whole circle is:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} A &amp; = &amp; \\pi \\times r^2 \\\\ &amp; = &amp; \\pi \\times 1.5^2 \\\\ &amp; = &amp; 2.25\\pi \\text{ square meters} \\end{array} [\/latex]<\/p>\n<p>Since the pattern has six segments, the area of one segment is:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{2.25\\pi}{6}=0.375\\pi\\approx 1.17 \\text{ square meters}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>translation symmetry<\/h3>\n<p><strong>Translation symmetry<\/strong>, also known as slide symmetry, involves an image or design being repeated in a straight line. It's like moving, or translating, an object without changing its orientation.<\/p>\n<p>&nbsp;<\/p>\n<center><img class=\"aligncenter wp-image-5067 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/53\/2025\/02\/13194511\/538px-Translation_of_a_set.svg_-1-300x191-1.png\" alt=\"Two identical shapes separated along a diagonal line\" width=\"300\" height=\"191\"><\/center><\/div>\n<\/section>\n<p>This type of symmetry is commonly used in wallpaper designs, textiles, and other works of art that involve repeated patterns.<\/p>\n<section class=\"textbox example\">\n<p>A wallpaper design features a pattern that repeats every [latex]0.4[\/latex] meters horizontally and every [latex]0.5[\/latex] meters vertically. Calculate the total number of patterns in a wall that is [latex]3[\/latex] meters high and [latex]2.8[\/latex] meters wide.<\/p>\n<p><br>\n[reveal-answer q=\"118114\"]Show Answer[\/reveal-answer]<br>\n[hidden-answer a=\"118114\"]The number of horizontal repeats:\n<\/p><center>[latex]\\frac{2.8}{0.4}=7[\/latex]<\/center>\n<p>The number of vertical repeats: \n<\/p><center>[latex]\\frac{3}{0.5}=6[\/latex]<\/center>\nTotal number of patterns:\n<center>[latex]7 \\times 6 = 42[\/latex]<\/center>\n[\/hidden-answer]\n<p>&nbsp;<\/p>\n<\/section>\n<p>&nbsp;<\/p>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Recognize the fundamental concepts of reflection, rotation, and translation symmetry<\/li>\n<\/ul>\n<\/section>\n<h2>Symmetry<\/h2>\n<p><strong>Symmetry<\/strong>, in its simplest form, refers to a sense of harmonious and aesthetically pleasing proportionality and balance. It&#8217;s a principle that is deeply rooted in nature, mathematics, art, design, and architecture. The basic idea behind symmetry is that if you were to draw a line through an object, shape, or design, one side would be a mirror image of the other.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5043\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/53\/2025\/02\/13194508\/Asymmetric_PSF.svg_.png\" alt=\"Two drawings of a tree are show, one with a line creating a symmetrical image and another with a line created an asymmetrical iamge\" width=\"500\" height=\"310\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>The line or plane that divides the shape, object or design into two equal and mirrored halves is called the <strong>axis of symmetry<\/strong>. This axis can be vertical, horizontal, or diagonal, depending on the orientation of the symmetrical shape.<\/p>\n<p>In visual arts, symmetry is used to create a sense of balance and harmony within a composition. Symmetrical compositions can be calming and stable, often conveying a sense of order and perfection.<\/p>\n<p>However, perfectly symmetrical compositions can also risk being seen as static or boring, which is why artists often balance symmetry with elements of asymmetry to create interest and dynamism. Breaking symmetry, while maintaining balance, can create visual interest and help guide the viewer&#8217;s eye to a focal point. This is often used in visual arts and design to create tension, emphasize certain elements, or convey movement and change.<\/p>\n<p>There are three primary types of symmetry &#8211; <strong>reflection<\/strong>, <strong>rotation<\/strong>, and <strong>translation symmetry<\/strong> &#8211; which are often used in different combinations to create aesthetically pleasing and balanced designs.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>reflection symmetry<\/h3>\n<p><strong>Reflection symmetry<\/strong>, also known as mirror or bilateral symmetry, occurs when one half of an image, shape or design is the mirror image of the other half. That is, one side reflects the other.<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-5047\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/53\/2025\/02\/13194509\/Symmetry-300x294-1.png\" alt=\"a set of four different shapes, three with reflection symmetry and one that is asymmetrical\" width=\"300\" height=\"294\" \/><\/div>\n<div style=\"text-align: center;\"><strong><span style=\"font-size: 10pt;\">Figures with the axes of symmetry drawn in. The figure with no axes is asymmetric.<\/span><\/strong><\/div>\n<\/div>\n<\/section>\n<p>Reflection symmetry is common in architecture and design, as it imparts a sense of balance and harmony. Artists may also use it to create emphasis on a particular aspect of their work. For instance, think of a butterfly. The right wing is a mirror image of the left wing, hence it has reflection symmetry. In architecture, many buildings and structures &#8211; from the Taj Mahal to modern skyscrapers &#8211; feature reflection symmetry.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>rotation symmetry<\/h3>\n<p><strong>Rotation symmetry<\/strong>, or radial symmetry, happens when a design or image can be rotated around a central point and still appear the same. An object&#8217;s degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<figure id=\"attachment_5049\" aria-describedby=\"caption-attachment-5049\" style=\"width: 150px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" class=\"size-thumbnail wp-image-5049\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/53\/2025\/02\/13194510\/683px-Finland_road_sign_166_1995%E2%80%932020.svg_-150x150-1.png\" alt=\"A triangular sign with three degrees of rotation symmetry\" width=\"150\" height=\"150\" \/><figcaption id=\"caption-attachment-5049\" class=\"wp-caption-text\">An example of three degrees of rotation symmetry.<\/figcaption><\/figure>\n<figure id=\"attachment_5050\" aria-describedby=\"caption-attachment-5050\" style=\"width: 150px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" class=\"size-thumbnail wp-image-5050\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/53\/2025\/02\/13194510\/Cyclic_symmetry_4-150x150-1.png\" alt=\"A drawing with four degrees of rotational symmetry\" width=\"150\" height=\"150\" \/><figcaption id=\"caption-attachment-5050\" class=\"wp-caption-text\">An example of four degrees of rotation symmetry.<\/figcaption><\/figure>\n<figure id=\"attachment_5051\" aria-describedby=\"caption-attachment-5051\" style=\"width: 150px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" class=\"size-thumbnail wp-image-5051\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/53\/2025\/02\/13194511\/599px-15crossings-decorative-knot.svg_-150x150-1.png\" alt=\"A drawing with five degrees of rotation symmetry\" width=\"150\" height=\"150\" \/><figcaption id=\"caption-attachment-5051\" class=\"wp-caption-text\">An example of five degrees of rotation symmetry.<\/figcaption><\/figure>\n<figure id=\"attachment_5052\" aria-describedby=\"caption-attachment-5052\" style=\"width: 150px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" class=\"size-thumbnail wp-image-5052\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/53\/2025\/02\/13194511\/459px-Olavsrose.svg_-150x150-1.png\" alt=\"A drawing with six degrees of rotational symmetry\" width=\"150\" height=\"150\" \/><figcaption id=\"caption-attachment-5052\" class=\"wp-caption-text\">An example of six degrees of rotation symmetry.<\/figcaption><\/figure>\n<\/section>\n<p>This type of symmetry is common in nature, such as in flowers and snowflakes, and is often used in design and art to create patterns that convey movement and dynamism.<\/p>\n<section class=\"textbox example\">\n<p>A designer is creating a circular tile pattern with sixfold rotation symmetry for a floor. If the radius of the circle is [latex]1.5[\/latex] meters, calculate the area of one segment (slice) of the pattern.<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q872000\">Show Answer<\/button><\/p>\n<div id=\"q872000\" class=\"hidden-answer\" style=\"display: none\">The area of the whole circle is:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} A & = & \\pi \\times r^2 \\\\ & = & \\pi \\times 1.5^2 \\\\ & = & 2.25\\pi \\text{ square meters} \\end{array}[\/latex]<\/p>\n<p>Since the pattern has six segments, the area of one segment is:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{2.25\\pi}{6}=0.375\\pi\\approx 1.17 \\text{ square meters}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>translation symmetry<\/h3>\n<p><strong>Translation symmetry<\/strong>, also known as slide symmetry, involves an image or design being repeated in a straight line. It&#8217;s like moving, or translating, an object without changing its orientation.<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5067 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/53\/2025\/02\/13194511\/538px-Translation_of_a_set.svg_-1-300x191-1.png\" alt=\"Two identical shapes separated along a diagonal line\" width=\"300\" height=\"191\" \/><\/div>\n<\/div>\n<\/section>\n<p>This type of symmetry is commonly used in wallpaper designs, textiles, and other works of art that involve repeated patterns.<\/p>\n<section class=\"textbox example\">\n<p>A wallpaper design features a pattern that repeats every [latex]0.4[\/latex] meters horizontally and every [latex]0.5[\/latex] meters vertically. Calculate the total number of patterns in a wall that is [latex]3[\/latex] meters high and [latex]2.8[\/latex] meters wide.<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q118114\">Show Answer<\/button><\/p>\n<div id=\"q118114\" class=\"hidden-answer\" style=\"display: none\">The number of horizontal repeats:\n<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{2.8}{0.4}=7[\/latex]<\/div>\n<p>The number of vertical repeats:\n<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{3}{0.5}=6[\/latex]<\/div>\n<p>Total number of patterns:<\/p>\n<div style=\"text-align: center;\">[latex]7 \\times 6 = 42[\/latex]<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/section>\n<p>&nbsp;<\/p>\n","protected":false},"author":6,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":450,"module-header":"","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/462"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/462\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/450"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/462\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=462"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=462"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=462"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=462"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}