{"id":155272,"date":"2022-03-26T00:27:43","date_gmt":"2022-03-25T18:57:43","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/line-integral-definition-and-examples-with-solutions\/"},"modified":"2022-12-26T23:16:29","modified_gmt":"2022-12-26T17:46:29","slug":"line-integral-definition-and-examples-with-solutions","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/line-integral\/","title":{"rendered":"Line Integral \u2013 Definition and Examples with Solutions"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/line-integral\/#What_is_a_Line_Integral\" title=\"What is a Line Integral?\">What is a Line Integral?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/line-integral\/#Line_Integral_Definition\" title=\"Line Integral Definition\">Line Integral Definition<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/line-integral\/#Line_Integral_of_the_Vector_Field\" title=\"Line Integral of the Vector Field\">Line Integral of the Vector Field<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/line-integral\/#Application_of_Line_Integral\" title=\"Application of Line Integral\">Application of Line Integral<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/line-integral\/#Step-by-Step_Guide_to_solving_Line_Integrals\" title=\"Step-by-Step Guide to solving Line Integrals\">Step-by-Step Guide to solving Line Integrals<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/line-integral\/#Difference_between_Line_Integrals_and_Definite_Integrals\" title=\"Difference between Line Integrals and Definite Integrals\">Difference between Line Integrals and Definite Integrals<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/line-integral\/#Tips_to_understand_Line_Integrals\" title=\"Tips to understand Line Integrals:\">Tips to understand Line Integrals:<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"What_is_a_Line_Integral\"><\/span>What is a Line Integral?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Line Integral \u2013 Definition: A <a href=\"https:\/\/infinitylearn.com\/surge\/maths\/line-integral\/\">line integral<\/a> is a calculation that determines the total length of a curve. The calculation is done by integrating the curve&#8217;s function along the curve.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-155271 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/line-integral-definition-and-examples-with-solutions.jpg\" alt=\"Line Integral \u2013 Definition and Examples with Solutions\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/line-integral-definition-and-examples-with-solutions.jpg?v=1648234658 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/line-integral-definition-and-examples-with-solutions-300x212.jpg?v=1648234658 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Line_Integral_Definition\"><\/span>Line Integral Definition<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The line integral of a function is the sum of the products of the function&#8217;s value at each point in the line segment between two points and the length of the line segment.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Line_Integral_of_the_Vector_Field\"><\/span>Line Integral of the Vector Field<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The line integral of the vector field is a measure of the total amount of work done by the vector field as a vector moves along a curve. line integral computed by taking the integral of the vector field over the entire curve.<\/p>\n<p>The line integral of a vector field also called a path integral.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Application_of_Line_Integral\"><\/span>Application of Line Integral<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Let us consider a simple example of a line integral. Suppose we want to calculate the line integral of the function along the curve<\/p>\n<ul>\n<li>We can use the fundamental theorem of calculus to calculate this line integral. We know that the derivative of is , so the line integral can written as<\/li>\n<li>can evaluate this line integral by integrating it over the curve . This gives<\/li>\n<li>We can also write this line integral in terms of the variables and :<\/li>\n<li>This line integral can interpreted as the amount of fluid (in units of ) that flows through the curve in a given time interval .<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"Step-by-Step_Guide_to_solving_Line_Integrals\"><\/span>Step-by-Step Guide to solving Line Integrals<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>This is a step-by-step guide on how to find the line integral of a function<\/p>\n<ol>\n<li>Identify the function and its derivatives<\/li>\n<li>Choose a path along which to integrate<\/li>\n<li>Evaluate the integral at each point on the path<\/li>\n<li>Combine the results to find the line integral<\/li>\n<\/ol>\n<h2><span class=\"ez-toc-section\" id=\"Difference_between_Line_Integrals_and_Definite_Integrals\"><\/span>Difference between Line Integrals and Definite Integrals<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li>The main difference between line integrals and definite integrals is that line integrals involve multiple integrals over a curve, while definite integrals involve a single integral over a given interval.<\/li>\n<li>Another difference is that line integrals always yield a result in terms of a physical quantity, such as displacement, velocity, or energy, while definite integrals can yield a variety of results, depending on the function integrated.<\/li>\n<li>Finally, line integrals are more difficult to compute than definite integrals, because they require evaluating the integrals over the entire curve, whereas definite integrals can evaluated by breaking the interval into smaller pieces and using the Fundamental Theorem of Calculus.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Tips_to_understand_Line_Integrals\"><\/span>Tips to understand Line Integrals:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ol>\n<li>Line integrals used to calculate the work done in moving a charge along a path in a electric field or the flux of a vector field through a surface.<\/li>\n<li>integrals are a way of calculating the accumulation of a vector field along a path.<\/li>\n<li>Line integrals can also used to calculate the potential energy of a system.<\/li>\n<li>integrals are a way of calculating the circulation of a vector field.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>What is a Line Integral? 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