{"id":2330,"date":"2022-02-25T18:01:11","date_gmt":"2022-02-25T09:01:11","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2330"},"modified":"2025-06-11T09:12:49","modified_gmt":"2025-06-11T00:12:49","slug":"2%e9%87%8d%e7%a9%8d%e5%88%86","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/%e5%a4%9a%e9%87%8d%e7%a9%8d%e5%88%86%ef%bc%9a%e5%a4%9a%e5%a4%89%e6%95%b0%e9%96%a2%e6%95%b0%e3%81%ae%e7%a9%8d%e5%88%86\/2%e9%87%8d%e7%a9%8d%e5%88%86\/","title":{"rendered":"2\u91cd\u7a4d\u5206"},"content":{"rendered":"<p>\u591a\u91cd\u7a4d\u5206\u3068\u3057\u3066\u6700\u3082\u7c21\u5358\u306a\uff0c2\u91cd\u7a4d\u5206 $\\displaystyle \\iint_D f(x, y) \\,dx \\,dy $ \u304c\u8868\u3059\u7acb\u4f53\u306e\u4f53\u7a4d\u3068\u306f&#8230;<br \/>\n<!--more--><\/p>\n<h3 id=\"yui_3_17_2_1_1645779329996_5716\">2\u91cd\u7a4d\u5206\u306e\u5b9a\u7fa9<\/h3>\n<div id=\"yui_3_17_2_1_1645779329996_5720\">1\u5909\u6570\u306e\u5834\u5408\u306e\u7a4d\u5206\uff0c\u4f8b\u3048\u3070 \\(\\displaystyle \\int_a^b f(x)\\,dx\\) \u306f\u4f55\u3092\u8868\u3057\u3066\u3044\u305f\u304b\u3068\u3044\u3046\u3068\uff0c\u66f2\u7dda \\( y = f(x)\\) \u3068 \\(y = 0\\)\uff08\\(x\\)\u8ef8\uff09\u3068\uff0c\\(x = a, \\ x = b\\) \u3067\u56f2\u307e\u308c\u305f\u90e8\u5206\u306e\u9762\u7a4d\u3092\u8868\u3057\u3066\u3044\u305f\u306e\u3067\u3042\u3063\u305f\u3002<\/div>\n<div id=\"yui_3_17_2_1_1645779329996_5721\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-8542\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/Area1.svg\" alt=\"\" width=\"640\" height=\"481\" \/><\/div>\n<div><\/div>\n<div>\u591a\u91cd\u7a4d\u5206\uff08\u591a\u5909\u6570\u95a2\u6570\u306e\u7a4d\u5206\uff09\u3068\u3057\u3066\u6700\u3082\u7c21\u5358\u306a\uff0c2\u5909\u6570\u306e\u7a4d\u5206\u3092<br \/>\n$$ \\iint_D f(x, y) \\,dx \\,dy $$<br \/>\n\u3068\u66f8\u304d\uff0c\u3053\u308c\u3092\u9818\u57df \\(D\\) \u3067\u306e2\u91cd\u7a4d\u5206\u3068\u547c\u3076\u3053\u3068\u306b\u3059\u308b\u3068\uff0c\u3053\u306e2\u91cd\u7a4d\u5206\u304c\u8868\u3057\u3066\u3044\u308b\u306e\u306f\uff0c\u66f2\u9762 \\(z = f(x, y)\\) \u3068 \\(z = 0\\)\uff08\\(xy\\) \u5e73\u9762\uff09\u306e\u9593\u3067\uff0c\u9818\u57df \\(D\\) \u306e\u4e0a\u306b\u3042\u308b\u90e8\u5206\u306e\u4f53\u7a4d $V$ \u3092\u8868\u3057\u3066\u3044\u308b\u3053\u3068\u306b\u306a\u308b\u3002<\/div>\n<div><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-8554 size-large\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/nijusekibun-vol.svg\" alt=\"\" width=\"640\" height=\"481\" \/><\/div>\n<div><\/div>\n<p>\u7279\u306b\uff0c\\(\\displaystyle \\iint_D \\,dx\\, dy \\) \u306f\uff08\u9ad8\u3055 \\(1\\) \u306e\u7acb\u4f53\u306e\u4f53\u7a4d\u3067\u3042\u308a\uff0c\u4f53\u7a4d\u3068\u306f\u5e95\u9762\u7a4d\u00d7\u9ad8\u3055\u3067\u3042\u308b\u304b\u3089\uff09\u9818\u57df \\(D\\) \u306e\u9762\u7a4d\u3092\u4e0e\u3048\u308b\u3002<\/p>\n<h3>\u9762\u7a4d\u5206\uff0c\u4f53\u7a4d\u7a4d\u5206<\/h3>\n<p>&#8230; \u3068\u4ee5\u4e0a\u306e\u3088\u3046\u306b\u8aac\u660e\u3059\u308b\u3068\uff0c1\u5909\u6570\u95a2\u6570\u306e\u7a4d\u5206\u304c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u9762\u7a4d<\/strong><\/span>\uff0c2\u5909\u6570\u95a2\u6570\u306e\u7a4d\u5206\u3064\u307e\u308a2\u91cd\u7a4d\u5206\u304c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u4f53\u7a4d<\/strong><\/span>\u3068\u77ed\u7d61\u7684\u306b\u8003\u3048\u3066\u3057\u307e\u3044\u305d\u3046\u306a\u306e\u3067\uff0c\u88dc\u8db3\u3002<\/p>\n<h4>\u9762\u7a4d\u306f\u9762\u7a4d\u5206\u304b\u3089<\/h4>\n<p>\u672c\u6765\u306f\uff0c$xy$ \u5e73\u9762\u4e0a\u306e\u3042\u308b\u9818\u57df $D$ \u306e\u9762\u7a4d $S$ \u306f<\/p>\n<p>$$ S \\equiv \\iint_D dx\\, dy$$<\/p>\n<p>\u306e\u3088\u3046\u306b\uff08\u88ab\u7a4d\u5206\u95a2\u6570\u304c $1$ \u306e\uff092\u91cd\u7a4d\u5206\uff08\u9762\u7a4d\u5206\uff09\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u3055\u3089\u306b\u306f\u4e00\u822c\u7684\u306b\u66f2\u9762\u4e0a\u306e\u3042\u308b\u9818\u57df $D$ \u306e\u5834\u5408\u306b\u306f\uff0c\uff08\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19 $x, y$ \u4ee5\u5916\u3082\u4f7f\u3046\u3060\u308d\u3046\u304b\u3089\uff09\u5fae\u5c0f\u9762\u7a4d\u8981\u7d20\u3092 $dx\\, dy \\rightarrow dS$ \u3068\u66f8\u3044\u3066<\/p>\n<p>$$ S = \\iint_D dS$$<\/p>\n<p>\u306a\u3069\u3068\u66f8\u304f\u3002<\/p>\n<p>\u9818\u57df $D$ \u304c\u7279\u306b $D: a \\leq x \\leq b, \\ 0 \\leq y \\leq f(x)$ \u3068\u66f8\u3051\u308b\u5834\u5408\u306b\u306f\uff0c\u5f8c\u3067\u8aac\u660e\u3059\u308b<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u7d2f\u6b21\u7a4d\u5206<\/strong><\/span>\u306e\u8981\u9818\u3067<\/p>\n<p>\\begin{eqnarray}<br \/>\nS &amp;=&amp; \\iint_D dx\\, dy \\\\<br \/>\n&amp;=&amp; \\int_a^b \\left\\{ \\int_0^{f(x)} dy\\right\\}dx\u00a0 \u00a0\\\\<br \/>\n&amp;=&amp; \\int_a^b \\bigl[y \\bigr]_0^{f(x)} \\ dx\u00a0 \\\\<br \/>\n&amp;=&amp; \\int_a^b f(x)\\,dx<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3068\u306a\u308b\u3002\u672c\u6765\uff0c\u9762\u7a4d\u3068\u3044\u3046\u3082\u306e\u306f\uff08\u88ab\u7a4d\u5206\u95a2\u6570\u304c $1$ \u306e\uff092\u91cd\u7a4d\u5206\uff08\u9762\u7a4d\u5206\uff09\u3067\u3042\u308a\uff0c\u9818\u57df $D$ \u304c\u3053\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u5834\u5408\u306b\u306f1\u5909\u6570\u95a2\u6570\u306e\u7a4d\u5206\u306b\u5e30\u7740\u3059\u308b\u3093\u3060\u3088\uff0c\u3068\u7406\u89e3\u3059\u308b\u3002<\/p>\n<h4>\u4f53\u7a4d\u306f\u4f53\u7a4d\u7a4d\u5206\u304b\u3089<\/h4>\n<p>\u540c\u69d8\u306b\u4f53\u7a4d\u306b\u3064\u3044\u3066\u3082\uff0c\u672c\u6765\u306f3\u6b21\u5143\u7a7a\u9593\u5185\u306e\u3042\u308b\u9818\u57df $D_3$ \u306e\u4f53\u7a4d $V$ \u306f<\/p>\n<p>$$ V \\equiv \\iiint_{D_3} dx\\, dy\\, dz$$<\/p>\n<p>\u306e\u3088\u3046\u306b\uff08\u88ab\u7a4d\u5206\u95a2\u6570\u304c $1$ \u306e\uff093\u91cd\u7a4d\u5206\uff08\u4f53\u7a4d\u7a4d\u5206\uff09\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u3055\u3089\u306b\u306f\u4e00\u822c\u7684\u306b\u306f\uff08\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19 $x, y, z$ \u4ee5\u5916\u3082\u4f7f\u3046\u3060\u308d\u3046\u304b\u3089\uff09\u5fae\u5c0f\u9762\u7a4d\u8981\u7d20\u3092 $dx\\, dy\\,dz \\rightarrow dV$ \u3068\u66f8\u3044\u3066<\/p>\n<p>$$ V = \\iiint_{D_3}\u00a0 dV$$<\/p>\n<p>\u306a\u3069\u3068\u66f8\u304f\u3002\u9818\u57df $D_3$ \u304c\u7279\u306b $D_3: D(x, y), \\ 0 \\leq z \\leq f(x, y)$ \u3068\u66f8\u3051\u308b\u5834\u5408\u306b\u306f\uff0c\u5f8c\u3067\u8aac\u660e\u3059\u308b<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u7d2f\u6b21\u7a4d\u5206<\/strong><\/span>\u306e\u8981\u9818\u3067<\/p>\n<p>\\begin{eqnarray}<br \/>\nV &amp;=&amp; \\iiint_{D_3} dx\\, dy\\, dz \\\\<br \/>\n&amp;=&amp; \\iint_D \\left\\{\\int_0^{f(x, y)} dz \\right\\} dx\\, dy\u00a0 \\\\<br \/>\n&amp;=&amp; \\iint_D \\bigl[z \\bigr]_0^{f(x, y)} \\ dx\\, dy\u00a0 \\\\<br \/>\n&amp;=&amp; \\iint_D f(x, y)\\\u00a0 dx\\, dy<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3068\u306a\u308b\u3002\u672c\u6765\uff0c\u4f53\u7a4d\u3068\u3044\u3046\u3082\u306e\u306f\uff08\u88ab\u7a4d\u5206\u95a2\u6570\u304c $1$ \u306e\uff093\u91cd\u7a4d\u5206\uff08\u4f53\u7a4d\u7a4d\u5206\uff09\u3067\u3042\u308a\uff0c\u9818\u57df $D_3$ \u304c\u3053\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u5834\u5408\u306b\u306f2\u5909\u6570\u95a2\u6570\u306e2\u91cd\u7a4d\u5206\uff08\u9762\u7a4d\u5206\uff09\u306b\u5e30\u7740\u3059\u308b\u3093\u3060\u3088\uff0c\u3068\u7406\u89e3\u3059\u308b\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u591a\u91cd\u7a4d\u5206\u3068\u3057\u3066\u6700\u3082\u7c21\u5358\u306a\uff0c2\u91cd\u7a4d\u5206 $\\displaystyle \\iint_D f(x, y) \\,dx \\,dy $ \u304c\u8868\u3059\u7acb\u4f53\u306e\u4f53\u7a4d\u3068\u306f&#8230;<\/p><p><a class=\"more-link btn\" href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/%e5%a4%9a%e9%87%8d%e7%a9%8d%e5%88%86%ef%bc%9a%e5%a4%9a%e5%a4%89%e6%95%b0%e9%96%a2%e6%95%b0%e3%81%ae%e7%a9%8d%e5%88%86\/2%e9%87%8d%e7%a9%8d%e5%88%86\/\">\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2228,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2330","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2330","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=2330"}],"version-history":[{"count":13,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2330\/revisions"}],"predecessor-version":[{"id":10484,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2330\/revisions\/10484"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2228"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2330"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}