{"id":130,"date":"2026-03-14T20:12:53","date_gmt":"2026-03-14T20:12:53","guid":{"rendered":"https:\/\/seonumber1.com\/calc\/?page_id=130"},"modified":"2026-03-19T20:27:37","modified_gmt":"2026-03-19T20:27:37","slug":"integral-calculator","status":"publish","type":"page","link":"https:\/\/seonumber1.com\/calc\/integral-calculator\/","title":{"rendered":"Integral Calculator"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"130\" class=\"elementor elementor-130\">\n\t\t\t\t<div class=\"elementor-element elementor-element-ce26ffd e-flex e-con-boxed e-con e-parent\" data-id=\"ce26ffd\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-d665114 elementor-widget elementor-widget-html\" data-id=\"d665114\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t\t<link href=\"https:\/\/fonts.googleapis.com\/css2?family=DM+Sans:wght@400;500;600;700&display=swap\" rel=\"stylesheet\">\r\n<style>*,*::before,*::after{box-sizing:border-box;margin:0;padding:0}.cw{font-family:'DM Sans',sans-serif;background:#f5f0e8;color:#1a2744;padding:40px 20px;max-width:720px;margin:0 auto}.cw h1{font-size:clamp(1.55rem,3vw,2rem);font-weight:700;text-align:center;margin-bottom:8px}.sub{font-size:.9rem;color:#718096;text-align:center;margin-bottom:28px;line-height:1.6}.tabs{display:flex;gap:6px;flex-wrap:wrap;margin-bottom:20px;justify-content:center}.tab{padding:7px 15px;border:1.5px solid #e2e8f0;border-radius:6px;font-family:inherit;font-size:.8rem;font-weight:600;cursor:pointer;background:#fff;color:#4a5568;transition:all .18s}.tab.on{background:#e8392a;color:#fff;border-color:#e8392a}.cc{background:#fff;border:1px solid #e2e8f0;border-radius:12px;padding:28px;margin-bottom:20px;box-shadow:0 2px 12px rgba(0,0,0,.06)}.cc 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li{margin-bottom:3px}@media(max-width:520px){.fr{grid-template-columns:1fr}}<\/style>\r\n<div class=\"cw\">\r\n  <h1>\u222b Integral Calculator<\/h1>\r\n  <p class=\"sub\">Calculate definite integrals numerically using Simpson's Rule, Trapezoidal Rule, and Midpoint Rule with step-by-step solutions.<\/p>\r\n  <div class=\"tabs\">\r\n    <button class=\"tab on\" onclick=\"sw(0)\">Simpson's Rule<\/button>\r\n    <button class=\"tab\" onclick=\"sw(1)\">Trapezoidal Rule<\/button>\r\n    <button class=\"tab\" onclick=\"sw(2)\">Midpoint Rule<\/button>\r\n    <button class=\"tab\" onclick=\"sw(3)\">Common Integrals<\/button>\r\n  <\/div>\r\n  <div class=\"cc\">\r\n    <div class=\"panel on\" id=\"p0\">\r\n      <h2>Definite Integral \u2014 Simpson's Rule<\/h2>\r\n      <div class=\"fr\">\r\n        <div class=\"fd\"><label>Function f(x)<\/label><select id=\"s_fx\"><option value=\"x2\">x\u00b2<\/option><option value=\"x3\">x\u00b3<\/option><option value=\"sinx\">sin(x)<\/option><option value=\"cosx\">cos(x)<\/option><option value=\"ex\">e\u02e3<\/option><option value=\"lnx\">ln(x)<\/option><option value=\"sqrtx\">\u221ax<\/option><option value=\"1x\">1\/x<\/option><\/select><\/div>\r\n        <div class=\"fd\"><label>Number of Intervals (even)<\/label><input type=\"number\" id=\"s_n\" placeholder=\"e.g. 8\" min=\"2\" max=\"1000\" step=\"2\" value=\"8\"><\/div>\r\n      <\/div>\r\n      <div class=\"fr\">\r\n        <div class=\"fd\"><label>Lower Limit a<\/label><input type=\"number\" id=\"s_a\" placeholder=\"e.g. 0\" step=\"any\"><\/div>\r\n        <div class=\"fd\"><label>Upper Limit b<\/label><input type=\"number\" id=\"s_b\" placeholder=\"e.g. 4\" step=\"any\"><\/div>\r\n      <\/div>\r\n      <button class=\"btn\" onclick=\"calcIntegral('s')\">Calculate Integral<\/button>\r\n      <div class=\"rb\" id=\"sr\">\r\n        <div class=\"rm\" id=\"s_res\">--<\/div>\r\n        <div class=\"rl\">\u222b f(x) dx \u2248 (Simpson's Rule)<\/div>\r\n        <div class=\"steps\" id=\"s_steps\"><\/div>\r\n      <\/div>\r\n    <\/div>\r\n    <div class=\"panel\" id=\"p1\">\r\n      <h2>Definite Integral \u2014 Trapezoidal Rule<\/h2>\r\n      <div class=\"fr\">\r\n        <div class=\"fd\"><label>Function f(x)<\/label><select id=\"t_fx\"><option value=\"x2\">x\u00b2<\/option><option value=\"x3\">x\u00b3<\/option><option value=\"sinx\">sin(x)<\/option><option value=\"cosx\">cos(x)<\/option><option value=\"ex\">e\u02e3<\/option><option value=\"lnx\">ln(x)<\/option><option value=\"sqrtx\">\u221ax<\/option><option value=\"1x\">1\/x<\/option><\/select><\/div>\r\n        <div class=\"fd\"><label>Number of Intervals n<\/label><input type=\"number\" id=\"t_n\" placeholder=\"e.g. 10\" min=\"2\" max=\"1000\" value=\"10\"><\/div>\r\n      <\/div>\r\n      <div class=\"fr\">\r\n        <div class=\"fd\"><label>Lower Limit a<\/label><input type=\"number\" id=\"t_a\" placeholder=\"e.g. 0\" step=\"any\"><\/div>\r\n        <div class=\"fd\"><label>Upper Limit b<\/label><input type=\"number\" id=\"t_b\" placeholder=\"e.g. 4\" step=\"any\"><\/div>\r\n      <\/div>\r\n      <button class=\"btn\" onclick=\"calcIntegral('t')\">Calculate Integral<\/button>\r\n      <div class=\"rb\" id=\"tr\">\r\n        <div class=\"rm\" id=\"t_res\">--<\/div>\r\n        <div class=\"rl\">\u222b f(x) dx \u2248 (Trapezoidal Rule)<\/div>\r\n        <div class=\"steps\" id=\"t_steps\"><\/div>\r\n      <\/div>\r\n    <\/div>\r\n    <div class=\"panel\" id=\"p2\">\r\n      <h2>Definite Integral \u2014 Midpoint Rule<\/h2>\r\n      <div class=\"fr\">\r\n        <div class=\"fd\"><label>Function f(x)<\/label><select id=\"m_fx\"><option value=\"x2\">x\u00b2<\/option><option value=\"x3\">x\u00b3<\/option><option value=\"sinx\">sin(x)<\/option><option value=\"cosx\">cos(x)<\/option><option value=\"ex\">e\u02e3<\/option><option value=\"lnx\">ln(x)<\/option><option value=\"sqrtx\">\u221ax<\/option><\/select><\/div>\r\n        <div class=\"fd\"><label>Number of Rectangles n<\/label><input type=\"number\" id=\"m_n\" placeholder=\"e.g. 10\" min=\"1\" max=\"1000\" value=\"10\"><\/div>\r\n      <\/div>\r\n      <div class=\"fr\">\r\n        <div class=\"fd\"><label>Lower Limit a<\/label><input type=\"number\" id=\"m_a\" placeholder=\"e.g. 0\" step=\"any\"><\/div>\r\n        <div class=\"fd\"><label>Upper Limit b<\/label><input type=\"number\" id=\"m_b\" placeholder=\"e.g. 4\" step=\"any\"><\/div>\r\n      <\/div>\r\n      <button class=\"btn\" onclick=\"calcIntegral('m')\">Calculate Integral<\/button>\r\n      <div class=\"rb\" id=\"mr\">\r\n        <div class=\"rm\" id=\"m_res\">--<\/div>\r\n        <div class=\"rl\">\u222b f(x) dx \u2248 (Midpoint Rule)<\/div>\r\n        <div class=\"steps\" id=\"m_steps\"><\/div>\r\n      <\/div>\r\n    <\/div>\r\n    <div class=\"panel\" id=\"p3\">\r\n      <h2>Common Integral Formulas Reference<\/h2>\r\n      <div style=\"font-size:.84rem;color:#4a5568;line-height:2\">\r\n        <div style=\"padding:6px 0;border-bottom:1px solid #f0eae0\"><span style=\"font-weight:700;color:#e8392a\">\u222b x\u207f dx<\/span> = x\u207f\u207a\u00b9 \/ (n+1) + C &nbsp;&nbsp; (n \u2260 \u22121)<\/div>\r\n        <div style=\"padding:6px 0;border-bottom:1px solid #f0eae0\"><span style=\"font-weight:700;color:#e8392a\">\u222b 1\/x dx<\/span> = ln|x| + C<\/div>\r\n        <div style=\"padding:6px 0;border-bottom:1px solid #f0eae0\"><span style=\"font-weight:700;color:#e8392a\">\u222b e\u02e3 dx<\/span> = e\u02e3 + C<\/div>\r\n        <div style=\"padding:6px 0;border-bottom:1px solid #f0eae0\"><span style=\"font-weight:700;color:#e8392a\">\u222b sin(x) dx<\/span> = \u2212cos(x) + C<\/div>\r\n        <div style=\"padding:6px 0;border-bottom:1px solid #f0eae0\"><span style=\"font-weight:700;color:#e8392a\">\u222b cos(x) dx<\/span> = sin(x) + C<\/div>\r\n        <div style=\"padding:6px 0;border-bottom:1px solid #f0eae0\"><span style=\"font-weight:700;color:#e8392a\">\u222b ln(x) dx<\/span> = x\u00b7ln(x) \u2212 x + C<\/div>\r\n        <div style=\"padding:6px 0\"><span style=\"font-weight:700;color:#e8392a\">\u222b \u221ax dx<\/span> = (2\/3)x^(3\/2) + C<\/div>\r\n      <\/div>\r\n    <\/div>\r\n  <\/div>\r\n  <div class=\"ib\"><h3>Accuracy Notes<\/h3><p>Simpson's Rule is generally most accurate (4th-order error). Trapezoidal is 2nd-order. More intervals (n) = higher accuracy for all methods.<\/p><ul><li>Simpson's error \u2248 \u2212(b\u2212a)\u2075\/(180n\u2074) \u00d7 f\u2074(\u03be)<\/li><li>Trapezoidal error \u2248 \u2212(b\u2212a)\u00b3\/(12n\u00b2) \u00d7 f\u2033(\u03be)<\/li><\/ul><\/div>\r\n<\/div>\r\n<script>\r\nfunction sw(i){document.querySelectorAll('.tab').forEach(function(t,j){t.classList.toggle('on',j===i)});document.querySelectorAll('.panel').forEach(function(p,j){p.classList.toggle('on',j===i)});}\r\nvar fns={x2:function(x){return x*x;},x3:function(x){return x*x*x;},sinx:Math.sin,cosx:Math.cos,ex:Math.exp,lnx:Math.log,sqrtx:Math.sqrt,'1x':function(x){return 1\/x;}};\r\nvar fnLabels={x2:'x\u00b2',x3:'x\u00b3',sinx:'sin(x)',cosx:'cos(x)',ex:'e\u02e3',lnx:'ln(x)',sqrtx:'\u221ax','1x':'1\/x'};\r\nfunction f4(v){return parseFloat(v.toFixed(6));}\r\nfunction calcIntegral(m){\r\n  var fx=document.getElementById(m+'_fx').value,n=+document.getElementById(m+'_n').value,a=+document.getElementById(m+'_a').value,b=+document.getElementById(m+'_b').value;\r\n  if(isNaN(a)||isNaN(b)||!n){alert('Fill all fields.');return;}\r\n  if(m==='s'&&n%2!==0){n++;document.getElementById('s_n').value=n;}\r\n  var fn=fns[fx],lbl=fnLabels[fx],h=(b-a)\/n,res=0,steps='';\r\n  if(m==='s'){\r\n    var vals=[];for(var i=0;i<=n;i++)vals.push(fn(a+i*h));\r\n    res=vals[0]+vals[n];\r\n    for(var i=1;i<n;i++)res+=(i%2===0?2:4)*vals[i];\r\n    res*=h\/3;\r\n    steps='<div class=\"step\"><span class=\"step-n\">Function:<\/span> f(x) = '+lbl+'<\/div><div class=\"step\"><span class=\"step-n\">Interval:<\/span> ['+a+', '+b+'], n='+n+', h='+(h.toFixed(4))+'<\/div><div class=\"step\"><span class=\"step-n\">Formula:<\/span> (h\/3)[f(a)+4f(x\u2081)+2f(x\u2082)+...+f(b)]<\/div><div class=\"step\"><span class=\"step-n\">f(a):<\/span> '+f4(vals[0])+'<\/div><div class=\"step\"><span class=\"step-n\">f(b):<\/span> '+f4(vals[n])+'<\/div><div class=\"step\"><span class=\"step-n\">Result:<\/span> \u2248 '+f4(res)+'<\/div>';\r\n  } else if(m==='t'){\r\n    var vals=[];for(var i=0;i<=n;i++)vals.push(fn(a+i*h));\r\n    res=(vals[0]+vals[n]);\r\n    for(var i=1;i<n;i++)res+=2*vals[i];\r\n    res*=h\/2;\r\n    steps='<div class=\"step\"><span class=\"step-n\">Function:<\/span> f(x) = '+lbl+'<\/div><div class=\"step\"><span class=\"step-n\">Interval:<\/span> ['+a+', '+b+'], n='+n+', h='+h.toFixed(4)+'<\/div><div class=\"step\"><span class=\"step-n\">Formula:<\/span> (h\/2)[f(x\u2080)+2f(x\u2081)+...+2f(x\u2099\u208b\u2081)+f(x\u2099)]<\/div><div class=\"step\"><span class=\"step-n\">f(a):<\/span> '+f4(vals[0])+'<\/div><div class=\"step\"><span class=\"step-n\">f(b):<\/span> '+f4(vals[n])+'<\/div><div class=\"step\"><span class=\"step-n\">Result:<\/span> \u2248 '+f4(res)+'<\/div>';\r\n  } else {\r\n    for(var i=0;i<n;i++){var mid=a+(i+0.5)*h;res+=fn(mid);}\r\n    res*=h;\r\n    steps='<div class=\"step\"><span class=\"step-n\">Function:<\/span> f(x) = '+lbl+'<\/div><div class=\"step\"><span class=\"step-n\">Interval:<\/span> ['+a+', '+b+'], n='+n+', h='+h.toFixed(4)+'<\/div><div class=\"step\"><span class=\"step-n\">Formula:<\/span> h \u00d7 \u03a3 f(midpoints)<\/div><div class=\"step\"><span class=\"step-n\">Result:<\/span> \u2248 '+f4(res)+'<\/div>';\r\n  }\r\n  document.getElementById(m+'_res').textContent='\u2248 '+f4(res);\r\n  document.getElementById(m+'_steps').innerHTML=steps;\r\n  document.getElementById(m+'r').classList.add('show');\r\n}\r\n<\/script>\r\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-cf33a47 e-flex e-con-boxed e-con e-parent\" data-id=\"cf33a47\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-6326446 elementor-widget elementor-widget-text-editor\" data-id=\"6326446\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<h3 class=\"text-xl font-semibold text-foreground mb-4 flex items-center\">How the\u00a0Integral Calculator\u00a0Works<\/h3><div class=\"space-y-6\"><div><h4 class=\"font-semibold mb-3 text-lg\">How Integration Works<\/h4><p class=\"text-sm mb-4 text-muted-foreground\">Integration finds the area under a curve by dividing it into small sections and summing their areas. This calculator uses numerical methods to approximate definite integrals when analytical solutions are difficult.<\/p><\/div><div><h4 class=\"font-semibold mb-2\">Mathematical Formulas:<\/h4><div class=\"space-y-3 text-sm\"><div class=\"bg-secondary\/10 p-3 rounded\"><strong>Simpson&#8217;s Rule:<\/strong><br \/>\u222b[a,b] f(x)dx \u2248 (h\/3)[f(x\u2080) + 4f(x\u2081) + 2f(x\u2082) + 4f(x\u2083) + &#8230; + f(x\u2099)]<br \/><em class=\"text-muted-foreground\">Most accurate for smooth functions<\/em><\/div><div class=\"bg-secondary\/10 p-3 rounded\"><strong>Trapezoidal Rule:<\/strong><br \/>\u222b[a,b] f(x)dx \u2248 (h\/2)[f(x\u2080) + 2f(x\u2081) + 2f(x\u2082) + &#8230; + f(x\u2099)]<br \/><em class=\"text-muted-foreground\">Good balance of accuracy and simplicity<\/em><\/div><div class=\"bg-secondary\/10 p-3 rounded\"><strong>Rectangular Rule:<\/strong><br \/>\u222b[a,b] f(x)dx \u2248 h[f(x\u2081) + f(x\u2082) + &#8230; + f(x\u2099)]<br \/><em class=\"text-muted-foreground\">Basic approximation method<\/em><\/div><\/div><\/div><div><h4 class=\"font-semibold mb-2\">Real-World Applications:<\/h4><ul class=\"list-disc ml-6 space-y-1 text-sm text-muted-foreground\"><li><strong>Physics:<\/strong>\u00a0Calculate work done by variable forces, center of mass<\/li><li><strong>Engineering:<\/strong>\u00a0Find areas of irregular shapes, fluid flow analysis<\/li><li><strong>Economics:<\/strong>\u00a0Consumer surplus, producer surplus calculations<\/li><li><strong>Statistics:<\/strong>\u00a0Probability density function areas, normal distribution<\/li><li><strong>Medicine:<\/strong>\u00a0Drug concentration over time, bioavailability studies<\/li><\/ul><\/div><div><h4 class=\"font-semibold mb-2\">Usage Recommendations:<\/h4><ul class=\"list-disc ml-6 space-y-1 text-sm text-muted-foreground\"><li>Use Simpson&#8217;s Rule for smooth, continuous functions (most accurate)<\/li><li>Choose Trapezoidal Rule for general-purpose calculations<\/li><li>Increase intervals (n) for higher accuracy, especially for oscillating functions<\/li><li>For functions like x\u00b2, x\u00b3, sin(x), cos(x), use at least 10-20 intervals<\/li><li>Verify results by comparing different methods when precision is critical<\/li><\/ul><\/div><\/div>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>\u222b Integral Calculator Calculate definite integrals numerically using Simpson&#8217;s Rule, Trapezoidal Rule, and Midpoint Rule with step-by-step solutions. Simpson&#8217;s Rule [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"elementor_header_footer","meta":{"site-sidebar-layout":"no-sidebar","site-content-layout":"","ast-site-content-layout":"full-width-container","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"disabled","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"class_list":["post-130","page","type-page","status-publish","hentry"],"_hostinger_reach_plugin_has_subscription_block":false,"_hostinger_reach_plugin_is_elementor":false,"_links":{"self":[{"href":"https:\/\/seonumber1.com\/calc\/wp-json\/wp\/v2\/pages\/130","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/seonumber1.com\/calc\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/seonumber1.com\/calc\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/seonumber1.com\/calc\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/seonumber1.com\/calc\/wp-json\/wp\/v2\/comments?post=130"}],"version-history":[{"count":7,"href":"https:\/\/seonumber1.com\/calc\/wp-json\/wp\/v2\/pages\/130\/revisions"}],"predecessor-version":[{"id":290,"href":"https:\/\/seonumber1.com\/calc\/wp-json\/wp\/v2\/pages\/130\/revisions\/290"}],"wp:attachment":[{"href":"https:\/\/seonumber1.com\/calc\/wp-json\/wp\/v2\/media?parent=130"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}