{"id":1343,"date":"2026-01-27T15:03:03","date_gmt":"2026-01-27T05:03:03","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1343"},"modified":"2026-01-27T15:03:03","modified_gmt":"2026-01-27T05:03:03","slug":"year12-math-3-3-3-integral-calculus","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1343","title":{"rendered":"Year12 MATH 3-3-3 integral calculus"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">In QLD Year 12 Mathematical Methods, Unit 3: Topic 3: <strong>Integrals<\/strong><mark>focuses on the concept of anti-differentiation (the reverse of differentiation) and its application to calculating areas<\/mark>. It serves as a foundational component of calculus in Senior Mathematics, bridging the gap between derivatives and total change.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Key components of Unit 3 Integral Calculus include:&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">1. Fundamental Concepts of Integration&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Anti-differentiation:<\/strong> Finding the original function from its derivative, including the addition of the constant of integration (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>+<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"text\/plain\">positive c<\/annotation><\/semantics><\/math>+\ud835\udc50).<\/li>\n\n\n\n<li><strong>Reverse Chain Rule:<\/strong> Techniques for integrating composite functions.<\/li>\n\n\n\n<li><strong>Integral Types:<\/strong> Understanding both indefinite integrals (finding the general formula) and definite integrals (evaluating between specific limits <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>a<\/mi><annotation encoding=\"text\/plain\">a<\/annotation><\/semantics><\/math>\ud835\udc4e and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>b<\/mi><annotation encoding=\"text\/plain\">b<\/annotation><\/semantics><\/math>\ud835\udc4f).<\/li>\n\n\n\n<li><strong>Integration by Recognition:<\/strong> A method where students use previously calculated derivatives to determine antiderivatives.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">2. Integration Techniques and Functions&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Polynomials:<\/strong> Basic integration rules.<\/li>\n\n\n\n<li><strong>Exponential Functions:<\/strong> Integrating <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><msup><mi>e<\/mi><mi>x<\/mi><\/msup><annotation encoding=\"text\/plain\">e to the x-th power<\/annotation><\/semantics><\/math>\ud835\udc52\ud835\udc65 and related exponential functions.<\/li>\n\n\n\n<li><strong>Logarithmic Functions:<\/strong> Integrating functions that result in natural logarithms (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>ln<\/mi><mo>|<\/mo><mi>x<\/mi><mo>|<\/mo><\/mrow><annotation encoding=\"text\/plain\">l n the absolute value of x end-absolute-value<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n<li><strong>Trigonometric Functions:<\/strong> Integrating <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>sin<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">sine x<\/annotation><\/semantics><\/math>sin(\ud835\udc65), <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>cos<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">cosine x<\/annotation><\/semantics><\/math>cos(\ud835\udc65), and related trigonometric forms.\u00a0<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">3. Applications of Integration&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Area Under a Curve:<\/strong> Calculating the area bounded by a function, the x-axis, and vertical lines.<\/li>\n\n\n\n<li><strong>Area Between Curves:<\/strong> Finding the area trapped between two distinct functions.<\/li>\n\n\n\n<li><strong>Total Change:<\/strong> Using integration to calculate the total change in a quantity from its rate of change.<\/li>\n\n\n\n<li><strong>Kinematics:<\/strong> Determining displacement and velocity from acceleration.<\/li>\n\n\n\n<li><strong>Trapezoidal Rule:<\/strong> A numerical method used for approximating definite integrals.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">This topic heavily emphasizes the link between differentiation and integration through the <strong>Fundamental Theorem of Calculus<\/strong>. Access to technology (such as CAS calculators) is assumed for calculating complex integrals.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">********************************************************************************************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">QLD 12\u5e74\u751f\u6570\u5b66\u6307\u5c0e\u6cd5\u306e\u30e6\u30cb\u30c3\u30c83\u300c\u30c8\u30d4\u30c3\u30af3\uff1a\u7a4d\u5206\u300d\u3067\u306f\u3001\u53cd\u5fae\u5206\uff08\u5fae\u5206\u306e\u9006\uff09\u306e\u6982\u5ff5\u3068\u3001\u9762\u7a4d\u8a08\u7b97\u3078\u306e\u5fdc\u7528\u306b\u7126\u70b9\u3092\u5f53\u3066\u3066\u3044\u307e\u3059\u3002\u3053\u308c\u306f\u4e0a\u7d1a\u6570\u5b66\u306b\u304a\u3051\u308b\u5fae\u7a4d\u5206\u306e\u57fa\u790e\u8981\u7d20\u3067\u3042\u308a\u3001\u5fae\u5206\u3068\u5168\u5909\u5316\u306e\u9593\u306e\u6a4b\u6e21\u3057\u3092\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30e6\u30cb\u30c3\u30c83\u300c\u7a4d\u5206\u5fae\u7a4d\u5206\u300d\u306e\u4e3b\u306a\u69cb\u6210\u8981\u7d20\u306f\u4ee5\u4e0b\u306e\u3068\u304a\u308a\u3067\u3059\u3002<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u7a4d\u5206\u306e\u57fa\u672c\u6982\u5ff5<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u53cd\u5fae\u5206\uff1a\u7a4d\u5206\u5b9a\u6570 (<math data-latex=\"+c\"><semantics><mrow><mo>+<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">+c<\/annotation><\/semantics><\/math>) \u306e\u52a0\u7b97\u3092\u542b\u3081\u3001\u5c0e\u95a2\u6570\u304b\u3089\u5143\u306e\u95a2\u6570\u3092\u6c42\u3081\u308b\u3002<br>\u9006\u9023\u9396\u5f8b\uff1a\u5408\u6210\u95a2\u6570\u306e\u7a4d\u5206\u624b\u6cd5\u3002<br>\u7a4d\u5206\u306e\u7a2e\u985e\uff1a\u4e0d\u5b9a\u7a4d\u5206\uff08\u4e00\u822c\u5f0f\u3092\u6c42\u3081\u308b\uff09\u3068\u5b9a\u7a4d\u5206\uff08\u7279\u5b9a\u306e\u6975\u9650 (<math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math>) \u3068 (<math data-latex=\"b\"><semantics><mi>b<\/mi><annotation encoding=\"application\/x-tex\">b<\/annotation><\/semantics><\/math>) \u306e\u9593\u306e\u8a55\u4fa1\uff09\u306e\u4e21\u65b9\u3092\u7406\u89e3\u3059\u308b\u3002<br>\u8a8d\u8b58\u306b\u3088\u308b\u7a4d\u5206\uff1a\u751f\u5f92\u304c\u4ee5\u524d\u306b\u8a08\u7b97\u3057\u305f\u5fae\u5206\u3092\u7528\u3044\u3066\u53cd\u5fae\u5206\u3092\u6c42\u3081\u308b\u65b9\u6cd5\u3002<\/p>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li>\u7a4d\u5206\u6280\u6cd5\u3068\u95a2\u6570<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u591a\u9805\u5f0f\uff1a\u57fa\u672c\u7684\u306a\u7a4d\u5206\u898f\u5247\u3002<br>\u6307\u6570\u95a2\u6570\uff1a(<math data-latex=\"e^{x}\"><semantics><msup><mi>e<\/mi><mi>x<\/mi><\/msup><annotation encoding=\"application\/x-tex\">e^{x}<\/annotation><\/semantics><\/math>) \u304a\u3088\u3073\u95a2\u9023\u3059\u308b\u6307\u6570\u95a2\u6570\u306e\u7a4d\u5206\u3002<br>\u5bfe\u6570\u95a2\u6570\uff1a\u81ea\u7136\u5bfe\u6570 (<math data-latex=\"\\ln |x|\"><semantics><mrow><mrow><mi>ln<\/mi><mo>\u2061<\/mo><\/mrow><mi>|<\/mi><mi>x<\/mi><mi>|<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\ln |x|<\/annotation><\/semantics><\/math>) \u3068\u306a\u308b\u95a2\u6570\u306e\u7a4d\u5206\u3002<br>\u4e09\u89d2\u95a2\u6570\uff1a(<math data-latex=\"\\sin (x)\"><semantics><mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\sin (x)<\/annotation><\/semantics><\/math>)\u3001(<math data-latex=\"\\cos (x)\"><semantics><mrow><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\cos (x)<\/annotation><\/semantics><\/math>)\u3001\u304a\u3088\u3073\u95a2\u9023\u3059\u308b\u4e09\u89d2\u95a2\u6570\u306e\u7a4d\u5206\u3002<\/p>\n\n\n\n<ol start=\"3\" class=\"wp-block-list\">\n<li>\u7a4d\u5206\u306e\u5fdc\u7528<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u66f2\u7dda\u4e0b\u306e\u9762\u7a4d\uff1a\u95a2\u6570\u3001x \u8ef8\u3001\u304a\u3088\u3073\u5782\u76f4\u7dda\u3067\u56f2\u307e\u308c\u305f\u9762\u7a4d\u3092\u8a08\u7b97\u3059\u308b\u3002<br>\u66f2\u7dda\u9593\u306e\u9762\u7a4d\uff1a2 \u3064\u306e\u7570\u306a\u308b\u95a2\u6570\u306b\u631f\u307e\u308c\u305f\u9762\u7a4d\u3092\u6c42\u3081\u308b\u3002<br>\u7dcf\u5909\u5316\uff1a\u7a4d\u5206\u3092\u7528\u3044\u3066\u3001\u3042\u308b\u91cf\u306e\u5909\u5316\u7387\u304b\u3089\u7dcf\u5909\u5316\u3092\u8a08\u7b97\u3059\u308b\u3002<br>\u904b\u52d5\u5b66\uff1a\u52a0\u901f\u5ea6\u304b\u3089\u5909\u4f4d\u3068\u901f\u5ea6\u3092\u6c42\u3081\u308b\u3002<br>\u53f0\u5f62\u5247\uff1a\u5b9a\u7a4d\u5206\u306e\u8fd1\u4f3c\u5024\u3092\u6c42\u3081\u308b\u6570\u5024\u7684\u624b\u6cd5\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u30c8\u30d4\u30c3\u30af\u3067\u306f\u3001\u5fae\u7a4d\u5206\u5b66\u306e\u57fa\u672c\u5b9a\u7406\u3092\u901a\u3057\u3066\u3001\u5fae\u5206\u3068\u7a4d\u5206\u306e\u95a2\u9023\u6027\u306b\u3064\u3044\u3066\u91cd\u70b9\u7684\u306b\u5b66\u3073\u307e\u3059\u3002\u8907\u7d20\u7a4d\u5206\u306e\u8a08\u7b97\u306b\u306f\u3001CAS\u8a08\u7b97\u6a5f\u306a\u3069\u306e\u30c6\u30af\u30ce\u30ed\u30b8\u30fc\u3092\u5229\u7528\u3067\u304d\u308b\u3053\u3068\u304c\u524d\u63d0\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In QLD Year 12 Mathematical Methods, Unit 3: Topic 3: Integralsfocuses on the concept of anti-differentiation (the reverse of differentiation) and its application to calculating areas. It serves as a foundational component of calculus in Senior Mathematics, bridging the gap between derivatives and total change.&nbsp; Key components of Unit 3 Integral Calculus include:&nbsp; 1. Fundamental [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1343","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1343","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1343"}],"version-history":[{"count":2,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1343\/revisions"}],"predecessor-version":[{"id":1378,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1343\/revisions\/1378"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1343"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1343"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1343"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}