{"id":9067,"date":"2024-07-03T11:45:30","date_gmt":"2024-07-03T02:45:30","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=9067"},"modified":"2025-06-11T10:04:56","modified_gmt":"2025-06-11T01:04:56","slug":"%e7%b4%af%e6%ac%a1%e7%a9%8d%e5%88%86%e3%81%ae%e4%be%8b%e9%a1%8c","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%e3%81%ae%e8%a8%88%e7%ae%97%ef%bc%9a%e7%b4%af%e6%ac%a1%e7%a9%8d%e5%88%86%e3%81%ae%e3%81%be%e3%81%a8%e3%82%81\/%e7%b4%af%e6%ac%a1%e7%a9%8d%e5%88%86%e3%81%ae%e4%be%8b%e9%a1%8c\/","title":{"rendered":"\u7d2f\u6b21\u7a4d\u5206\u306e\u4f8b\u984c"},"content":{"rendered":"

<\/p>\n

\u4f8b\u984c<\/h3>\n

$\\displaystyle \\iint_D (2 + x)\\, dx\\, dy, \\quad D: x^2 + y^2 \\leq 1$<\/p>\n

\u9818\u57df $D$ \u306f\u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3057\u305f\u534a\u5f84 $1$ \u306e\u5186\u306e\u5185\u90e8\u3067\u3042\u308b\u304b\u3089\uff0c<\/p>\n

$$D: -1 \\leq x \\leq 1, \\quad -\\sqrt{1 -x^2} \\leq y \\leq \\sqrt{1 -x^2}$$<\/p>\n

\u3057\u305f\u304c\u3063\u3066\uff0c\u7d2f\u6b21\u7a4d\u5206\u306e\u8981\u9818\u3067<\/p>\n

\\begin{eqnarray}
\n\\iint_D (2 + x) dx\\, dy &=& \\int_{-1}^1 dx \\int_{-\\sqrt{1 -x^2}}^{\\sqrt{1 -x^2}} dy \\, (2+x) \\\\
\n&=& \\int_{-1}^1 dx\\\u00a0 (2 + x)\\, \\biggl[\\,y\\,\\biggr]_{-\\sqrt{1 -x^2}}^{\\sqrt{1 -x^2}} \\\\
\n&=& \\int_{-1}^1 dx \\, (4 + 2 x) \\sqrt{1 -x^2} \\\\
\n&=& 4 \\int_{-1}^1 \\sqrt{1 -x^2}\\, dx + \\int_{-1}^1 2 x \\sqrt{1 -x^2}\\, dx \\\\
\n&=& \\cdots \\\\
\n&=& 2 \\pi
\n\\end{eqnarray}<\/p>\n

\uff08\u5b66\u751f\u8af8\u541b\u306f\u3061\u3083\u3093\u3068\u5b9a\u7a4d\u5206\u306e\u90e8\u5206\u3092\u7701\u7565\u305b\u305a\u306b\u8a08\u7b97\u904e\u7a0b\u3092\u660e\u793a\u3057\u3066\u8a08\u7b97\u3059\u308b\u306e\u3067\u3059\u3088\u3002\uff09<\/p>\n

\u3053\u306e\u7acb\u4f53\u306f\uff0c$z$ \u8ef8\u3092\u4e2d\u5fc3\u8ef8\u3068\u3057\u305f\u5e95\u9762\u304c\u534a\u5f84 $1$ \u306e\u5186\u3068\uff0c$z=2$ \u3092\u542b\u3080\u5e73\u9762 $z = f(x, y) = 2 + x$ \u3067\u533a\u5207\u3089\u308c\u305f\u7acb\u4f53\u3067\u3042\u308b\u304b\u3089…<\/p>\n

\"\"<\/p>\n

\u5e95\u9762\u304c\u534a\u5f84 $1$ \u306e\u5186\u3067\u3042\u308b\u5186\u67f1\u3092\uff0c$z=2$ \u3092\u542b\u3080\u5e73\u9762 $z = f(x, y) = 2 + x$ \u3067\u659c\u3081\u306b\u5207\u3063\u305f\u3082\u306e\u306e\u4e0b\u90e8\u306e\u4f53\u7a4d\u306b\u76f8\u5f53\u3059\u308b\u3002<\/p>\n

\"\"<\/p>\n

\u3064\u307e\u308a\uff0c\u5e95\u9762\u7a4d\u304c\u534a\u5f84 $r=1$ \u306e\u5186\u306e\u9762\u7a4d $\\pi\\,r^2 = \\pi$\uff0c\u9ad8\u3055\u304c $h = 4$ \u306e\u5186\u67f1\u306e\u4f53\u7a4d\u306e\u534a\u5206\u3067\u3042\u308b\u304b\u3089…<\/p>\n

\"\"<\/p>\n

$$ V = \\pi\\,r^2 \\times h \\times \\frac{1}{2} = 2 \\pi$$<\/p>\n

\u3068\uff0c\u78ba\u304b\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n

\"\"<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":2334,"menu_order":10,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-9067","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\/9067","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=9067"}],"version-history":[{"count":23,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/9067\/revisions"}],"predecessor-version":[{"id":10486,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/9067\/revisions\/10486"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2334"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=9067"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}