{"id":2158,"date":"2022-02-22T15:18:15","date_gmt":"2022-02-22T06:18:15","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2158"},"modified":"2024-06-13T10:32:18","modified_gmt":"2024-06-13T01:32:18","slug":"%e7%a9%8d%e5%88%86%ef%bc%9a%e3%81%84%e3%81%8f%e3%81%a4%e3%81%8b%e3%81%ae%e5%bf%9c%e7%94%a8","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%a6b\/%e7%a9%8d%e5%88%86%ef%bc%9a%e3%81%84%e3%81%8f%e3%81%a4%e3%81%8b%e3%81%ae%e5%bf%9c%e7%94%a8\/","title":{"rendered":"\u7a4d\u5206\uff1a\u3044\u304f\u3064\u304b\u306e\u5fdc\u7528"},"content":{"rendered":"

<\/p>\n

\u9762\u7a4d<\/h3>\n

\"\"<\/p>\n

\\(y = f(x) \\ (>0), \\ y = 0\\)\uff08\\(x\\) \u8ef8\uff09\u304a\u3088\u3073 \\(x = a, \\ x =\u00a0 b\\ (> a)\\) \u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d \\(S\\) \u306f
$$S = \\int_a^b f(x)\\, dx$$ \u3067\u3042\u3063\u305f\u3002<\/p>\n

\"\"<\/p>\n

\u540c\u69d8\u306b\uff0c\\(y = f(x), \\ y = g(x)\\) (\\( f(x) > g(x)\\) )\u304a\u3088\u3073 \\(x = a, \\ x =\u00a0 b\\ (> a)\\) \u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u306f<\/p>\n

$$S = \\int_a^b \\left\\{f(x) -g(x)\\right\\} \\, dx$$<\/p>\n

\u66f2\u7dda\u306e\u9577\u3055<\/h3>\n

\"\"<\/p>\n

\\(y = f(x)\\) \u306e \\((x, y)\\) \u304b\u3089 \\((x+dx, y+dy)\\) \u307e\u3067\u306e\uff08\u5fae\u5c0f\uff09\u9577\u3055 \\(d\\ell \\) \u306f<\/p>\n

$$d\\ell = \\sqrt{dx^2 + dy^2} = \\sqrt{1 + \\left(\\frac{dy}{dx}\\right)^2} \\,dx$$ \u3067\u3042\u308b\u304b\u3089\uff0c\\((a, f(a))\\)\u00a0 \u304b\u3089 \\((b, f(b))\\)\u00a0 \u307e\u3067\u306e\u66f2\u7dda\u306e\u9577\u3055\u306f
$$ L = \\int_a^b d\\ell =\\int_a^b \\sqrt{1 + \\left(\\frac{dy}{dx}\\right)^2} \\,dx $$<\/p>\n

\u66f2\u7dda\u304c $x = x(t), \\ y = y(t)$ \u306e\u3088\u3046\u306b\u5a92\u4ecb\u5909\u6570\u8868\u793a\u3055\u308c\u3066\u3044\u308b\u5834\u5408\u306f\uff0c<\/p>\n

$$d\\ell = \\sqrt{dx^2 + dy^2} = \\sqrt{\\left(\\frac{dx}{dt}\\right)^2 + \\left(\\frac{dy}{dt}\\right)^2} \\,dt$$<\/p>\n

\u3067\u3042\u308b\u304b\u3089 $(x(t_1), y(t_1))$ \u304b\u3089 $(x(t_2), y(t_2))$ \u307e\u3067\u306e\u66f2\u7dda\u306e\u9577\u3055\u306f<\/p>\n

$$ L = \\int_{t_1}^{t_2} d\\ell =\\int_{t_1}^{t_2} \\sqrt{\\left(\\frac{dx}{dt}\\right)^2 + \\left(\\frac{dy}{dt}\\right)^2} \\,dt $$<\/p>\n

\u56de\u8ee2\u4f53\u306e\u8868\u9762\u7a4d<\/h3>\n

\"\"<\/p>\n

\"\"<\/p>\n

\u95a2\u6570 \\(y = f(x)\\) \u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b\u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u56de\u8ee2\u4f53\u306e\u5074\u9762\u306e\u9762\u7a4d \\(S\\) \u306f\uff0c\u4e0a\u56f3\u306e\u3088\u3046\u306a\u5fae\u5c0f\u9762\u7a4d \\(dS\\) \u304c\u5e45 \\(d\\ell\\)\uff0c\u5468\u9577 \\(2\\pi y\\) \u306e\u30ea\u30dc\u30f3\u72b6\u3067\u3042\u308b\u3053\u3068\u304b\u3089<\/p>\n

$$ S = \\int_a^b dS = \\int_a^b 2\\pi y \\, d\\ell = \\int_a^b 2\\pi y \\sqrt{1 + \\left(\\frac{dy}{dx}\\right)^2}\\,dx$$<\/p>\n

\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d<\/h3>\n

\"\"<\/p>\n

\"\"<\/p>\n

\u95a2\u6570 \\(y = f(x)\\) \u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b\u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d \\(V\\) \u306f\uff0c\u8584\u3044\u8f2a\u5207\u308a\u306b\u3057\u305f\u90e8\u5206\u306e\u5fae\u5c0f\u4f53\u7a4d \\(dV\\) \u304c \\(\\pi y^2 dx\\) \u3067\u3042\u308b\u3053\u3068\u304b\u3089<\/p>\n

$$ V = \\int_a^b dV = \\int_a^b \\pi y^2 \\,dx$$<\/p>\n

\u5ff5\u306e\u305f\u3081\uff0c$dV$ \u306f\u5207\u308a\u53e3\u306e\u8868\u9762\u7a4d $\\pi y^2$ \u306b\u539a\u307f $dx$ \u3092\u304b\u3051\u308b\uff0c\u3064\u307e\u308a $dV = \\pi y^2 \\,dx$\u3002$d\\ell$ \u306f\u539a\u307f\u3067\u306f\u306a\u3044\u3002\u539a\u307f\u3068\u306f\u5207\u308a\u53e3\u306b\u5782\u76f4\u306a\u9577\u3055\u3060\u304b\u3089\uff0c$d\\ell$ \u3067\u306f\u306a\u304f\uff0c$dx$\u3002<\/p>\n

\u4f8b\u984c<\/h3>\n

\u5186\u306e\u9762\u7a4d<\/h4>\n

\u534a\u5f84 \\(r\\) \u306e\u5186\u306e\u9762\u7a4d\u3002\u5186\u306e\u65b9\u7a0b\u5f0f \\(x^2 + y^2 = r^2\\) \u3088\u308a \\(y = \\pm \\sqrt{r^2 -x^2}\\)\u3002\u4e0a\u534a\u5206\u306e\u9762\u7a4d\uff0c\u3064\u307e\u308a \\(y = +\\sqrt{r^2-x^2}\\) \u3068 \\(x\\) \u8ef8\u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u3092\u6c42\u3081\u30662\u500d\u3059\u308c\u3070\u3088\u3044\u304b\u3089\uff0c<\/p>\n

\\begin{eqnarray}
S &=& 2 \\int_{-r}^r \\sqrt{r^2-x^2} \\,dx \\\\
&& \\Bigl( x = r\\sin t, \\ dx = r\\cos t\\, dt, \\ -r \\leq x \\leq r \\ \\Rightarrow\\ -\\frac{\\pi}{2} \\leq t \\leq \\frac{\\pi}{2}\u00a0 \\Bigr)\\\\
&=& 2 \\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}} \\sqrt{r^2 -r^2 \\sin^2 t} \\cdot r \\cos t\\, dt\\\\
&=& 2r^2 \\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}} \\cos^2 t\\, dt\\\\
&=& 2r^2 \\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}} \\frac{1 + \\cos 2 t}{2} \\, dt\\\\
&=& r^2 \\Bigl[ x + \\frac{\\sin 2 t}{2} \\Bigr]_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}}\\\\
&=& \\pi r^2
\\end{eqnarray}<\/p>\n

\u5186\u5468<\/h4>\n

\u534a\u5f84 \\(r\\) \u306e\u5186\u306e\u5186\u5468\u3002\u5186\u306e\u65b9\u7a0b\u5f0f \\(x^2 + y^2 = r^2\\) \u3088\u308a \\(y = \\pm \\sqrt{r^2 -x^2}\\)\u3002\u4e0a\u534a\u5206\u306e\u5186\u5468\u3092\u6c42\u3081\u30662\u500d\u3059\u308c\u3070\u3088\u3044\u3002<\/p>\n

$$y\uff1d\\sqrt{r^2 -x^2}, \\ \\ \\frac{dy}{dx} = \\frac{1}{2} (r^2 -x^2)^{-\\frac{1}{2}}\\cdot(-2 x) = -\\frac{x}{\\sqrt{r^2 -x^2}}$$
$$\\sqrt{1+ \\left(\\frac{dy}{dx}\\right)^2} = \\sqrt{1 + \\frac{x^2}{r^2 -x^2}} = \\frac{r}{\\sqrt{r^2-x^2}}$$<\/p>\n

\\begin{eqnarray}
L &=& 2 \\int_{-r}^r \\sqrt{1+ \\left(\\frac{dy}{dx}\\right)^2}\\,x \\\\
&=& 2\\int_{-r}^r \\frac{r}{\\sqrt{r^2-x^2}}\\,x \\\\
&=& 2 r \\int_{-r}^r \\frac{1}{\\sqrt{1 -\\left(\\frac{x}{r}\\right)^2}} \\,\\frac{dx}{r} \\\\
&& \\Bigl( \\frac{x}{r} = t, \\ \\frac{dx}{r} = dt, \\ -r \\leq x \\leq r \\ \\Rightarrow\\ -1 \\leq t \\leq 1\\Bigr)\\\\
&=& 2 r \\int_{-1}^1 \\frac{1}{\\sqrt{1-t^2}} \\,dt \\\\
&=& 2r \\Bigl[ \\sin^{-1} t \\Bigr]_{-1}^1 \\\\
&=& 2r \\left( \\frac{\\pi}{2} -\\left(-\\frac{\\pi}{2}\\right) \\right) \\\\
&=& 2 \\pi r
\\end{eqnarray}<\/p>\n

\u3042\u308b\u3044\u306f\uff0c\u5186\u306e\u65b9\u7a0b\u5f0f \\(x^2 + y^2 = r^2\\) \u304c<\/p>\n

$$x = r \\cos\\theta, \\quad y = r \\sin\\theta$$<\/p>\n

\u306e\u3088\u3046\u306b $\\theta$ \u306b\u3088\u3063\u3066\u5a92\u4ecb\u5909\u6570\u8868\u793a\u3055\u308c\u308b\u3068\u601d\u3046\u3068<\/p>\n

\\begin{eqnarray}
\nL &=& \\int_0^{2\\pi} \\sqrt{\\left(\\frac{dx}{d\\theta}\\right)^2 + \\left(\\frac{dy}{d\\theta}\\right)^2} \\,d\\theta\\\\
\n&=& \\int_0^{2\\pi} r \\,d\\theta \\\\
\n&=& 2 \\pi \\,r
\n\\end{eqnarray}<\/p>\n

\u7403\u306e\u8868\u9762\u7a4d<\/h4>\n

\\( y = \\sqrt{r^2 -x^2}\\) \u3092\\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b\u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u534a\u5f84 \\(r\\) \u306e\u7403\u306e\u8868\u9762\u7a4d\u306f
\\begin{eqnarray}
S &=& \\int_{-r}^r 2\\pi y \\sqrt{1+ \\left(\\frac{dy}{dx}\\right)^2}\\,dx \\\\
&=& \\int_{-r}^r 2\\pi \\cdot\\sqrt{r^2 -x^2}\\cdot \\frac{r}{\\sqrt{r^2-x^2}}\\,dx \\\\
&=&2\\pi r\\Bigl[ x \\Bigr]_{-r}^r\\\\
&=&4\\pi r^2
\\end{eqnarray}<\/p>\n

\u7403\u306e\u4f53\u7a4d<\/h4>\n

\\( y = \\sqrt{r^2 -x^2}\\) \u3092\\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b\u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u534a\u5f84 \\(r\\) \u306e\u7403\u306e\u4f53\u7a4d\u306f<\/p>\n

\\begin{eqnarray}
V &=& \\int_{-r}^r \\pi y^2 \\, dx \\\\
&=& \\int_{-r}^r \\pi (r^2 -x^2)\\, dx \\\\
&=& 2 \\int_{0}^r \\pi (r^2 -x^2)\\, dx \\\\
&=& 2 \\pi \\Bigl[ r^2 x -\\frac{x^3}{3}\u00a0 \\Bigr]_{0}^r\\\\
&=& \\frac{4 \\pi}{3} r^3
\\end{eqnarray}<\/p>\n

\u3068\u3044\u3046\u308f\u3051\u3067\uff0c\u9ad8\u6821\u6642\u4ee3\u306a\u3089\u516c\u5f0f\u3068\u3057\u3066\u6697\u8a18\u3057\u3066\u304a\u3051\u3068\u3067\u3082\u8a00\u308f\u308c\u305f\u3067\u3042\u308d\u3046\uff0c\u5186\u5468\uff0c\u5186\u306e\u9762\u7a4d\uff0c\u7403\u306e\u8868\u9762\u7a4d\uff0c\u7403\u306e\u4f53\u7a4d\u304c\u5168\u3066\u76f4\u63a5\u7a4d\u5206\u3059\u308b\u3053\u3068\u3067\u6c42\u3081\u3089\u308c\u305f\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":2068,"menu_order":32,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2158","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\/2158","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=2158"}],"version-history":[{"count":11,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2158\/revisions"}],"predecessor-version":[{"id":8896,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2158\/revisions\/8896"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2068"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2158"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}