{"id":2135,"date":"2023-04-02T10:05:09","date_gmt":"2023-04-02T02:05:09","guid":{"rendered":"https:\/\/www.appblog.cn\/?p=2135"},"modified":"2023-04-05T20:00:16","modified_gmt":"2023-04-05T12:00:16","slug":"foundation-of-advanced-mathematics-definition-and-properties-of-definite-integral","status":"publish","type":"post","link":"https:\/\/www.appblog.cn\/index.php\/2023\/04\/02\/foundation-of-advanced-mathematics-definition-and-properties-of-definite-integral\/","title":{"rendered":"\u9ad8\u7b49\u6570\u5b66\u57fa\u7840\uff1a\u5b9a\u79ef\u5206\u7684\u5b9a\u4e49\u4e0e\u6027\u8d28"},"content":{"rendered":"

\u5b9a\u79ef\u5206\u5b9a\u4e49<\/h2>\n

\u8bbe\u51fd\u6570$f(x)$\u5b9a\u4e49\u5728$[a,b]$\u4e0a\uff0c\u82e5\u5bf9$[a,b]$\u7684\u4efb\u4e00\u79cd\u5206\u6cd5$a=x_0<x_1<x_2<\u00b7\u00b7\u00b7<x_n=b$\uff0c\u4ee4$\\Delta x_i=xi-x<\/em>{i-1}$\uff0c\u4efb\u53d6$\\xi_i \\in [xi,x<\/em>{i-1}]$\uff0c\u53ea\u8981$\\lambda = \\max_{1\u2264i\u2264n}\\{\\Delta xi\\} \\to 0$\u65f6$\\sum<\/em>{i=1}^n f(\\xi_i)\\Delta x_i$\u603b\u8d8b\u4e8e\u786e\u5b9a\u7684\u6781\u9650$I$\uff0c\u5219\u79f0\u6b64\u6781\u9650$I$\u4e3a\u51fd\u6570$f(x)$\u5728\u533a\u95f4$[a,b]$\u4e0a\u7684\u5b9a\u79ef\u5206<\/strong>\uff0c\u8bb0\u4f5c$\\int_a^b f(x)dx$\uff0c\u5373<\/p>\n

<\/p>\n

$$
\n\\inta^b f(x)dx = \\lim \\limits<\/em>{\\lambda \\to 0} \\sum_{i=1}^n f(\\xi_i)\\Delta x_i
\n$$<\/p>\n

\u6b64\u65f6\u79f0$f(x)$\u5728$[a,b]$\u4e0a\u53ef\u79ef<\/strong>\u3002<\/p>\n

\u5b9a\u79ef\u5206\u7684\u51e0\u4f55\u610f\u4e49<\/h2>\n

\u66f2\u8fb9\u68af\u5f62\u9762\u79ef<\/p>\n

\u53ef\u79ef\u7684\u5145\u5206\u6761\u4ef6<\/h2>\n

\u5b9a\u74061\uff1a\u82e5\u51fd\u6570$f(x)$\u5728$[a,b]$\u4e0a\u8fde\u7eed\uff0c\u5219$f(x)$\u5728$[a,b]$\u53ef\u79ef
\n\u5b9a\u74062\uff1a\u82e5\u51fd\u6570$f(x)$\u5728$[a,b]$\u4e0a\u6709\u754c\uff0c\u4e14\u53ea\u6709\u6709\u9650\u4e2a\u95f4\u65ad\u70b9\uff0c\u5219$f(x)$\u5728$[a,b]$\u53ef\u79ef<\/p>\n

\u4f8b1\uff1a\u5229\u7528\u5b9a\u4e49\u8ba1\u7b97\u5b9a\u79ef\u5206$\\int_0^1 x^2 dx$<\/p>\n

\u89e3\uff1a\u5c06$[0,1]$\u8fdb\u884c$n$\u7b49\u5206\uff0c\u5206\u70b9\u4e3a$x_i=\\frac{i}{n} \\, (i=0,1,\u00b7\u00b7\u00b7,n)$<\/p>\n

\u53d6$\\xi_i = \\frac{i}{n}, \\Delta x_i = \\frac{1}{n} \\, (i=0,1,\u00b7\u00b7\u00b7,n)$\uff0c\u5219<\/p>\n

$$
\nf(\\xi_i)\\Delta x_i = \\xi_i^2\\Delta x_i = \\frac{i^2}{n^3}
\n$$<\/p>\n

$$
\n\\begin{align}
\n\\sum_{i=1}^n f(\\xi_i)\\Delta xi &= \\frac{1}{n^3} \\sum<\/em>{i=1}^n i^2 \\
\n&= \\frac{1}{n^3}\u00b7\\frac{1}{6}n(n+1)(2n+1) \\
\n&= \\frac{1}{6}(1+\\frac{1}{n})(2+\\frac{1}{n})
\n\\end{align}
\n$$<\/p>\n

\u8fd9\u91cc\u7528\u5230\u516c\u5f0f<\/p>\n

$$
\n\\sum_{i=1}^n i^2 = \\frac{1}{6}n(n+1)(2n+1)
\n$$<\/p>\n

$$
\n\\begin{align}
\n\\int0^1 x^2dx &= \\lim \\limits<\/em>{\\lambda \\to 0} \\sum_{i=1}^n \\xi_i^2\\Delta xi \\
\n&= \\lim \\limits<\/em>{n \\to \u221e} \\frac{1}{6}(1+\\frac{1}{n})(2+\\frac{1}{n}) \\
\n&= \\frac{1}{3}
\n\\end{align}
\n$$<\/p>\n

\u4f8b2\uff1a\u7528\u5b9a\u79ef\u5206\u8868\u793a\u4e0b\u5217\u6781\u9650<\/p>\n

$$
\n\\lim \\limits{n \\to \u221e} \\frac{1}{n} \\sum<\/em>{i=1}^n \\sqrt{1+\\frac{i}{n}} = \\lim \\limits{n \\to \u221e} \\sum<\/em>{i=1}^n \\sqrt{1+\\frac{i}{n}}\u00b7\\frac{1}{n} = \\int_0^1 \\sqrt{1+x}dx
\n$$<\/p>\n

$$
\n\\lim \\limits{n \\to \u221e} \\frac{1^p+2^p+\u00b7\u00b7\u00b7+n^p}{n^{p+1}} = \\lim \\limits<\/em>{n \\to \u221e} \\sum_{i=1}^n (\\frac{i}{n})^p\u00b7\\frac{1}{n} = \\int_0^1 x^pdx
\n$$<\/p>\n

\u5b9a\u79ef\u5206\u7684\u6027\u8d28<\/h2>\n

(1) $\\int_a^b f(x)dx = -\\int_b^a f(x)dx$<\/p>\n

(2) $\\int_a^a f(x)dx = 0$<\/p>\n

(3) $\\int_a^b dx = b-a$<\/p>\n

(4) $\\int_a^b kf(x)dx = k\\int_a^b f(x)dx$<\/p>\n

(5) $\\int_a^b [f(x) \u00b1 g(x)]dx = \\int_a^b f(x)dx \u00b1 \\int_a^b g(x)dx$<\/p>\n

(6) $\\int_a^b f(x)dx = \\int_a^c f(x)dx + \\int_c^b f(x)dx$<\/p>\n

(7) \u82e5\u5728$[a,b]$\u4e0a$f(x) \u2265 0$\uff0c\u5219$\\int_a^b f(x)dx \u2265 0$<\/p>\n

(8) \u82e5\u5728$[a,b]$\u4e0a$f(x) \u2264 g(x)$\uff0c\u5219$\\int_a^b f(x)dx \u2264 \\int_a^b g(x)dx$<\/p>\n

(9) $|\\int_a^b f(x)dx| \u2264 \\int_a^b |f(x)|dx$<\/p>\n

(10) \u8bbe$M=max{[a,b]}f(x),m=min<\/em>{[a,b]}f(x)$\uff0c\u5219$m(b-a) \u2264 \\int_a^b f(x)dx \u2264 M(b-a) \\, (a<b)$<\/p>\n

\u4f8b3\uff1a\u8bd5\u8bc1$1 \u2264 \\int_0^{\\frac{\\pi}{2}} \\frac{\\sin x}{x} dx \u2264 \\frac{\\pi}{2}$<\/p>\n

\u8bc1\uff1a\u8bbe$f(x) = \\frac{\\sin x}{x}$\uff0c\u5219\u5728$(0,\\frac{\\pi}{2})$\u4e0a\uff0c\u6709<\/p>\n

$f'(x) = \\frac{x\\cos x – \\sin x}{x^2} = \\frac{\\cos x}{x^2}(x-\\tan x) < 0$<\/p>\n

$\\therefore f(\\frac{\\pi}{2}) < f(x) < f(0)$<\/p>\n

\u5373$\\frac{2}{\\pi} < f(x) < 1, \\, x \\in (0,\\frac{\\pi}{2})$<\/p>\n

\u6545$\\int_0^{\\frac{\\pi}{2}} \\frac{2}{\\pi} dx \u2264 \\int_0^{\\frac{\\pi}{2}} f(x) dx \u2264 \\int_0^{\\frac{\\pi}{2}} 1 dx$<\/p>\n

\u5373$1 \u2264 \\int_0^{\\frac{\\pi}{2}} \\frac{\\sin x}{x} dx \u2264 \\frac{\\pi}{2}$<\/p>\n

\u79ef\u5206\u4e2d\u503c\u5b9a\u7406<\/h2>\n

\u82e5$f(x)\\in C[a,b]$\uff0c\u5219\u81f3\u5c11\u5b58\u5728\u4e00\u70b9$\\xi \\in [a,b]$\uff0c\u4f7f<\/p>\n

$$
\n\\int_a^b f(x) dx = f(\\xi)(b-a)
\n$$<\/p>\n

$$
\n\\frac{\\inta^b f(x) dx}{b-a} = \\frac{1}{b-a} \\lim \\limits<\/em>{n \\to \u221e} \\sum_{i=1}^n f(\\xii)\u00b7\\frac{b-a}{n} = \\lim \\limits<\/em>{n \\to \u221e} \\frac{1}{n} \\sum_{i=1}^n f(\\xi_i)
\n$$<\/p>\n

\u6545\u79ef\u5206\u4e2d\u503c\u5b9a\u7406\u662f\u6709\u9650\u4e2a\u6570\u7684\u5e73\u5747\u503c\u6982\u5ff5\u7684\u63a8\u5e7f\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u5b9a\u79ef\u5206\u5b9a\u4e49 \u8bbe\u51fd\u6570$f(x)$\u5b9a\u4e49\u5728$[a,b]$\u4e0a\uff0c\u82e5\u5bf9$[a,b]$\u7684\u4efb\u4e00\u79cd\u5206\u6cd5$a=x_0<x_1 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[509],"tags":[],"class_list":["post-2135","post","type-post","status-publish","format-standard","hentry","category-mathematics-fundamentals"],"_links":{"self":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2135","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/comments?post=2135"}],"version-history":[{"count":0,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2135\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2135"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2135"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2135"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}