{"id":559,"date":"2022-09-13T21:18:51","date_gmt":"2022-09-13T13:18:51","guid":{"rendered":"https:\/\/damakey.com\/?p=559"},"modified":"2023-05-24T15:59:26","modified_gmt":"2023-05-24T07:59:26","slug":"%e6%95%b8%e5%ad%b8%e5%a5%b3%e5%ad%a9-%e7%a9%8d%e5%88%86%e7%af%87","status":"publish","type":"post","link":"https:\/\/damakey.com\/?p=559","title":{"rendered":"\u6578\u5b78\u5973\u5b69 \u7a4d\u5206\u7bc7"},"content":{"rendered":"

<\/a>
<\/span><\/p>\n


\u8907\u7fd2\u4e00\u4e0b\u4e0d\u5b9a\u7a4d\u5206\uff0c\u4e0d\u91cd\u65b0\u770b\u4e00\u4e0b\uff0c\u9084\u771f\u7684\u9023\u6700\u57fa\u672c\u7684\u904b\u7b97\u90fd\u9084\u7d66\u8001\u5e2b\u4e86\u5462\uff01<\/span><\/p>\n

<\/span><\/p>\n

\u548c\u7684\u7a4d\u5206\u7b49\u65bc\u7a4d\u5206\u7684\u548c\u3002<\/span><\/p>\n

<\/span><\/p>\n

∫(a*f(x)+b*g(x))dx=a*∫f(x)dx+b*∫g(x)dx<\/span><\/p>\n

<\/span><\/p>\n

\u5169\u500b\u51fd\u6578\u4e58\u7a4d\u7684\u7a4d\u5206\uff0c\u5247\u4e0d\u7b49\u65bc\u5169\u500b\u51fd\u6578\u7a4d\u5206\u7684\u4e58\u7a4d\uff0c\u8981\u901a\u904e\u8f49\u63db\u624d\u53ef\u80fd\u7b97\u51fa\u4f86\u3002\u9019\u6bd4\u8f03\u4e0d\u76f4\u63a5\u3002<\/span><\/p>\n

<\/span><\/p>\n

\u8b6c\u5982\uff0c\u600e\u9ebc\u8a08\u7b97<\/span>∫xe^xdx  <\/span><\/span>\u9019\u500b\u7a4d\u5206\u5f0f\u5462\uff1f<\/span><\/p>\n

<\/span><\/p>\n

\u4e00\u6642\u662f\u4ee4\u4eba\u60f3\u4e0d\u51fa\u4f86\uff0c\u5230\u5e95\u662f\u54ea\u4e00\u500b\u51fd\u6578\u7684\u5fae\u5206\u662f<\/span>xe^x<\/span>\u3002<\/span><\/p>\n

<\/span><\/p>\n

\u5148\u8003\u616e\u5169\u500b\u51fd\u6578<\/span> f(x) <\/span>\u548c<\/span> g(x) <\/span>\u76f8\u4e58\u4e4b\u5f8c\u7684\u5fae\u5206\uff1a<\/span><\/p>\n

<\/span><\/p>\n

(f(x)g(x))’=f’(x)g(x)+f(x)g’(x)<\/span><\/p>\n

<\/span><\/p>\n

\u5169\u908a\u90fd\u53d6\u7a4d\u5206<\/span><\/p>\n

<\/span><\/p>\n

∫((f(x)g(x))’= ∫f’(x)g(x)+ ∫f(x)g’(x)<\/span><\/p>\n

<\/span><\/p>\n

f(x)g(x)+C= ∫f’(x)g(x)+ ∫f(x)g’(x) <\/span>\u79fb\u9805\u4e00\u4e0b<\/span><\/p>\n

<\/span><\/p>\n

∫f’(x)g(x)= f(x)g(x)- ∫f(x)g’(x)+C<\/span><\/p>\n

<\/span><\/p>\n

\u4ee4<\/span> f(x)=e^x; g(x)=x<\/span><\/p>\n

<\/span><\/p>\n

∫x(e^x)’dx=xe^x-∫(x)’e^xdx+C<\/span><\/p>\n

<\/span><\/p>\n

∫xe^xdx= xe^x-∫e^xdx+C<\/span><\/p>\n

<\/span><\/p>\n

∫xe^xdx= xe^x-e^x+C<\/span><\/p>\n

<\/span><\/p>\n

\u9019\u88cf\u7528\u5230\u4e86\u6307\u6578\u51fd\u6578<\/span>e^x<\/span>\u5fae\u5206\u4e4b\u5f8c\u9084\u662f<\/span>e^x<\/span>\u9019\u500b\u6027\u8cea\u3002<\/span><\/p>\n

<\/span><\/p>\n

\u7df4\u7fd2\u984c<\/span><\/p>\n

<\/span><\/p>\n

\u6c42\u89e3\uff1a<\/span> ∫(2x+3x^2+4x^3)dx<\/span><\/p>\n

=x^2+x^3+x^4+C<\/span><\/p>\n

<\/span><\/p>\n

\u6c42\u89e3\uff1a<\/span> ∫(x^2+e^x)<\/span><\/p>\n

=(1\/3)x^3+e^x+C<\/span><\/p>\n

<\/span><\/p>\n

\u6c42\u89e3\uff1a<\/span> ∫(n+1)!x^ndx<\/span><\/p>\n

=n!x^(n+1)+C<\/span><\/p>\n

<\/span><\/p>\n

\u6c42\u89e3\uff1a<\/span> ∫(12x^2+34e^x+56sinx)dx<\/span><\/p>\n

=4x^3+34e^x-56cosx+C<\/span><\/p>\n

<\/span><\/p>\n

\u6c42\u89e3\uff1a<\/span>∫xcosxdx<\/span><\/p>\n

<\/span><\/p>\n

∫(xsinx)’= ∫x(sinx)’dx+∫sinxdx+C<\/span><\/p>\n

xsinx= ∫xcosxdx-cosx+C<\/span><\/p>\n

<\/span><\/p>\n

∫xcosxdx=xsinx+cosx+C<\/span><\/p>\n

<\/span><\/p>\n

\u6c42\u89e3\uff1a<\/span>∫(x^2+x+1)e^xdx<\/span><\/p>\n

= ∫x^2e^xdx+ ∫xe^xdx+∫e^xdx+C<\/span><\/p>\n

<\/span><\/p>\n

\u5176\u4e2d<\/span>∫(x^2e^x)’dx= ∫x^2e^xdx+2∫xe^x+C <\/span>\u79fb\u9805\u4ee3\u5165\u4e0a\u5f0f<\/span><\/p>\n

<\/span><\/p>\n

=x^2e^x-2∫xe^x +∫xe^xdx+∫e^xdx+C<\/span><\/p>\n

= x^2e^x-∫xe^xdx+ e^x+C<\/span><\/p>\n

<\/span><\/p>\n

∫(xe^x)’dx= ∫xe^xdx+∫e^x+C= ∫xe^xdx+e^x+C <\/span>\u79fb\u9805\u4ee3\u5165\u4e0a\u5f0f<\/span><\/p>\n

<\/span><\/p>\n

=x^2e^x-xe^x+2e^x+C<\/span><\/p>\n

=(x^2-x+2)e^x+C<\/span><\/p>\n

<\/span><\/p>\n

\u6c42\u89e3\uff1a<\/span>∫ (1\/0!x^0+1\/1!x^1+1\/2!x^2+…+1\/n!x^n+…)dx<\/span><\/p>\n

=1\/1!x+1\/2!x2+…+1\/(n+1!)x^(n+1)+…<\/span><\/p>\n

= (1\/0!x^0+1\/1!x^1+1\/2!x^2+…+1\/n!x^n+…))-(1\/0!x^0)<\/span><\/p>\n

<\/span><\/p>\n

\u800c\u5176\u5be6<\/span>1\/0!x^0+1\/1!x^1+1\/2!x^2+…+1\/n!x^n+…)=e^x<\/span><\/p>\n

<\/span><\/p>\n

\u6240\u4ee5\u4e0a\u5f0f\u7684\u7a4d\u5206\u5c31\u662f<\/span>∫e^x=e^x+C<\/span>\u4e86<\/span><\/p>\n

<\/span><\/p>\n

\u7a4d\u5206\u5728\u5e7e\u4f55\u4e0a\u9762\u7684\u610f\u7fa9\uff0c\u7c21\u55ae\u8b1b\u5c31\u662f\u8a08\u7b97\u51fd\u6578\u4e0b\u7684\u9762\u7a4d\u3002\u800c\u5728\u6975\u5ea7\u6a19\u4e0a\uff0c\u7528\u4f86\u8a08\u7b97\u9310\u9ad4\u548c\u7403\u9ad4\u7684\u7684\u9762\u7a4d\u3001\u9ad4\u7a4d\uff0c\u4e5f\u90fd\u5f88\u65b9\u4fbf\u3002<\/span><\/p>\n

<\/span><\/p>\n

\u9310\u9ad4<\/span>(<\/span>\u5074<\/span>)<\/span>\u9762\u7a4d<\/span><\/p>\n

<\/span><\/p>\n

\u5047\u8a2d\u5713\u9310\u9ad8\u5ea6\u70ba<\/span>h<\/span>\uff0c\u5e95\u90e8\u7684\u534a\u5f91\u70ba<\/span>R<\/span>\uff0c\u5728\u9ad8\u70ba<\/span>x<\/span>\u7684\u5730\u65b9\u6a6b\u5207\u51fa\u4e00\u584a\u5c0f\u5713\uff0c\u539a\u5ea6<\/span>dx<\/span>\uff0c\u90a3\u9ebc\u90a3\u500b\u5c0f\u5713\u7684\u534a\u5f91\u5247\u70ba<\/span>R*(h-x)\/h<\/span>\uff0c\u90a3\u500b\u5c0f\u5713\u7684\u5916\u5074\u9762\u7a4d\u70ba<\/span>2π(R*(h-x)\/h)dx<\/span>\uff0c\u6240\u4ee5\u9310\u9ad4\u7684\u5074\u9762\u5c31\u5f9e<\/span>x=0 <\/span>\u5230<\/span>h <\/span>\u505a\u4e00\u500b\u4ee5\u4e0b\u7684\u7a4d\u5206\uff1a<\/span><\/p>\n

<\/span><\/p>\n

∫2π(R*(h-x)\/h)dx =2π(R\/h)*(hx-1\/2x^2)= πRh<\/span><\/p>\n

<\/span><\/p>\n

\u9310\u9ad4\u9ad4\u7a4d<\/span><\/p>\n

<\/span><\/p>\n

\u5047\u8a2d\u5713\u9310\u9ad8\u5ea6\u70ba<\/span>h<\/span>\uff0c\u5e95\u90e8\u7684\u534a\u5f91\u70ba<\/span>R<\/span>\uff0c\u5728\u9ad8\u70ba<\/span>x<\/span>\u7684\u5730\u65b9\u6a6b\u5207\u51fa\u4e00\u584a\u5c0f\u5713\uff0c\u539a\u5ea6<\/span>dx<\/span>\uff0c\u90a3\u9ebc\u90a3\u500b\u5c0f\u5713\u7684\u534a\u5f91\u5247\u70ba<\/span>R*(h-x)\/h<\/span>\uff0c\u90a3\u500b\u5c0f\u5713\u7684\u5916\u5074\u9762\u7a4d\u70ba<\/span>π(R*(h-x)\/h)^2dx<\/span>\uff0c\u6240\u4ee5\u9310\u9ad4\u7684\u5074\u9762\u5c31\u5f9e<\/span>x=0 <\/span>\u5230<\/span>h <\/span>\u505a\u4e00\u500b\u4ee5\u4e0b\u7684\u7a4d\u5206\uff1a<\/span><\/p>\n

<\/span><\/p>\n

∫π(R*(h-x)\/h)^2dx= π(R\/h)^2∫(h^2-2hx+x^2)= (1\/3)πR^2*h<\/span><\/p>\n

<\/span><\/p>\n

\u9019\u4ee3\u8868\u628a\u5713\u5468\u5f9e\u9ad8\u5ea6<\/span>x=0<\/span>\u958b\u59cb\uff0c\u4e00\u76f4\u5f80\u4e0a\u5806\u758a\uff0c\u76f4\u5230<\/span>x<\/span>\u9054\u5230\u9310\u9ad4\u7e3d\u9ad8\u5ea6<\/span> h<\/span>\u70ba\u6b62\uff0c\u5e95\u9762\u534a\u5f91\u70ba<\/span>R<\/span>\uff0c\u9ad8\u70ba<\/span>h<\/span>\u7684\u5713\u690e\u9ad4\u659c\u9762\u7684\u7e3d\u8868\u9762\u7a4d\u3002<\/span><\/p>\n

<\/span><\/p>\n

\u7403\u9ad4\u8868\u9762\u7a4d<\/span><\/p>\n

<\/span><\/p>\n

\u81f3\u65bc\u7403\u9ad4\u7684\u8868\u9762\u7a4d\uff0c\u60f3\u50cf\u4e00\u500b\u534a\u5f91\u70ba<\/span>R<\/span>\u7684\u7403\u9ad4\uff0c\u5f9e\u4e2d\u9593\u5207\u6210\u4e0a\u4e0b\u5169\u534a\uff0c\u5148\u7b97\u4e0a\u534a\u90e8\u3002\u5047\u8a2d\u628a\u4e0a\u534a\u90e8\u7684\u7403\u9ad4\u6a6b\u5207\u7121\u6578\u7247\uff0c\u5176\u4e2d\u4e00\u7247\u7684\u5e73\u9762\u534a\u5f91\u70ba<\/span>r<\/span>\uff0c\u5f9e\u7403\u5fc3\u5230\u9019\u500b\u534a\u5f91\u70ba<\/span>r<\/span>\u7684\u5713\u5468\u4e0a\u7684\u90a3\u689d\u7dda\u548c\u5e73\u9762\u7684\u89d2\u5ea6\u70ba<\/span>θ<\/span>\uff0c\u90a3\u9ebc<\/span>r<\/span>\u5c31\u662f<\/span>Rcosθ<\/span>\uff0c\u800c\u5728\u534a\u5f91\u70ba<\/span>r<\/span>\u7684\u90a3\u500b\u5c0f\u74b0\u7684\u539a\u5ea6\u662f<\/span>Rdθ<\/span>\uff0c\u90a3\u4e00\u5c0f\u5708\u7684\u9762\u7a4d\u5c31\u662f<\/span>2πr*Rdθ<\/span>\uff0c\u7b97\u7403\u4e0a\u534a\u908a\u7684\u8868\u9762\u7a4d\u5c31\u662f\u628a<\/span>2πr*Rdθ<\/span>\uff0c\u7531<\/span>θ=0<\/span>\u7a4d\u5206\u5230<\/span>θ=π\/2<\/span><\/p>\n

<\/span><\/p>\n

∫2πr*Rdθ= 2πr*R^2∫cosθ= 2πr*R^2(sin(π\/2)-sin(0))= 2πr*R^2 <\/span>\uff0c\u9019\u53ea\u662f\u534a\u500b\u7403<\/span><\/p>\n

<\/span><\/p>\n

\u6240\u4ee5\uff0c\u7403\u9ad4\u7684\u8868\u9762\u7a4d\u662f<\/span>4πr*R^2<\/span>\u3002<\/span><\/p>\n

<\/span><\/p>\n

\u7403\u9ad4\u9ad4\u7a4d<\/span><\/p>\n

<\/span><\/p>\n

\u5728\u8a08\u7b97\u7403\u9ad4\u9ad4\u7a4d\uff0c\u4e5f\u662f\u628a\u7403\u9ad4\u6a6b\u5207\u6210\u7d30\u8584\u7247\uff0c\u7136\u5f8c\u7d2f\u7a4d\u8d77\u4f86\u3002\u552f\u4e00\u8981\u6ce8\u610f\u7684\u662f\u7d30\u8584\u7247\u7684\u9ad8\u5ea6\u548c\u7403\u9762\u4e0a\u7684\u5f27\u8ddd<\/span>Rdθ<\/span>\u8981\u4e58\u4e0a<\/span>cosθ<\/span>\u4f86\u8a08\u7b97\u5176\u9ad8\u5ea6\u3002<\/span><\/p>\n

<\/span><\/p>\n

∫π(R*cosθ)^2*Rdθcosθ<\/span><\/p>\n

=πR^3∫cosθ^3dθ<\/span><\/p>\n

= πR^3*(sinθ-(sinθ ^3)\/3)  <\/span><\/span>\uff5c\u5176\u4e2d<\/span>θ =0 <\/span>\u5230<\/span>π\/2 <\/span><\/p>\n

= πR^3*<\/span>\uff08<\/span>1-0)-(1-0)\/3<\/span><\/p>\n

=(2\/3)πR^3 <\/span>\u9019\u53ea\u662f\u534a\u7403\u7684\u9ad4\u7a4d<\/span><\/p>\n

<\/span><\/p>\n

\u6545\u534a\u5f91\u70ba<\/span>R<\/span>\u7684\u5713\u7403\u9ad4\u7a4d\u70ba<\/span>(4\/3)πR^3<\/span><\/p>\n

<\/span><\/p>\n

P.S. <\/span>\u9918\u5f26\u4e09\u6b21\u65b9\u7528\u7f6e\u63db\u6cd5\u505a\u7a4d\u5206\uff1a<\/span><\/p>\n

<\/span><\/p>\n

∫cosθ^3dθ=∫(1-sinθ^2}* cosθdθ)  <\/span><\/p>\n

\u4ee4<\/span>u=sinθ du= cosθdθ<\/span><\/p>\n

∫(1-sinθ^2)cosθdθ)=∫(1-u^2)du=u-(1\/3)*u^3+C <\/span>\u628a<\/span>u=sinθ <\/span>\u4ee3\u56de\u53bb<\/span><\/p>\n

∫cosθ^3dθ= sinθ-(1\/3) sinθ^3   <\/span>(θ <\/span>\u7531<\/span>0 <\/span>\u7a4d\u5206\u5230<\/span>π\/2)<\/span><\/p>\n

=(1-0)-(1\/3)*(1-0)^3=2\/3<\/span><\/p>\n

<\/span><\/p>\n

(<\/span>\u4e0d\u67e5\u8cc7\u6599\uff0c\u9084\u771f\u7684\u5b8c\u5168\u5fd8\u8a18\u600e\u9ebc\u505a\u4e86\u5462\uff01<\/span>)<\/span><\/p>\n

<\/span><\/p>\n

\u9019\u4e9b\u90fd\u7b97\u662f\u5fae\u7a4d\u5206\u76f8\u5c0d\u6bd4\u8f03\u76f4\u89c0\u7684\u904b\u7528\uff0c\u5728\u8a08\u7b97\u7269\u9ad4\u7684\u9762\u7a4d\u548c\u9ad4\u7a4d\uff0c\u7528\u7a4d\u5206\u5c31\u662f\u758a\u52a0\u8d77\u4f86\u7684\u6982\u5ff5\uff0c\u5176\u5be6\u4e5f\u6c92\u6709\u60f3\u50cf\u4e2d\u7684\u90a3\u9ebc\u96e3\u61c2\u3002\u53ea\u662f\u5982\u679c\u727d\u6d89\u5230\u4e09\u89d2\u51fd\u6578\u7684\u7a4d\u5206\uff0c\u90a3\u5c31\u9700\u8981\u4e00\u9ede\u6280\u5de7\u3002\u9084\u7d66\u8001\u5e2b\u7684\uff0c\u73fe\u5728\u627e\u56de\u4e86\u4e00\u4e9b\u3002<\/span><\/p>\n

<\/span><\/p>\n

*<\/span>\uff1a\u300a\u6578\u5b78\u5973\u5b69<\/span> <\/span>\u7a4d\u5206\u7bc7\u300b\uff0c\u7d50\u57ce\u6d69<\/span> <\/span>\u8457\uff0c\u885e\u5bae\u7d18<\/span> <\/span>\u8b6f<\/span><\/p>\n

<\/span><\/p>\n

2022\/9\/6 <\/span>\u6578\u5b78\u5973\u5b69<\/span> <\/span>\u7a4d\u5206\u7bc7<\/span> Damakey<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

\u8907\u7fd2\u4e00\u4e0b\u4e0d\u5b9a\u7a4d\u5206\uff0c\u4e0d\u91cd\u65b0\u770b\u4e00\u4e0b\uff0c\u9084\u771f\u7684\u9023\u6700\u57fa\u672c\u7684\u904b\u7b97\u90fd\u9084\u7d66\u8001…<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"content-type":"","footnotes":""},"categories":[3],"tags":[],"class_list":["post-559","post","type-post","status-publish","format-standard","hentry","category-3"],"_links":{"self":[{"href":"https:\/\/damakey.com\/index.php?rest_route=\/wp\/v2\/posts\/559","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/damakey.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/damakey.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/damakey.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/damakey.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=559"}],"version-history":[{"count":1,"href":"https:\/\/damakey.com\/index.php?rest_route=\/wp\/v2\/posts\/559\/revisions"}],"predecessor-version":[{"id":19554,"href":"https:\/\/damakey.com\/index.php?rest_route=\/wp\/v2\/posts\/559\/revisions\/19554"}],"wp:attachment":[{"href":"https:\/\/damakey.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=559"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/damakey.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=559"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/damakey.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=559"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}