{"id":214,"date":"2023-08-05T01:28:35","date_gmt":"2023-08-05T01:28:35","guid":{"rendered":"https:\/\/rumusmatematika.id\/?p=214"},"modified":"2023-11-17T06:58:15","modified_gmt":"2023-11-17T06:58:15","slug":"volume-benda-putar","status":"publish","type":"post","link":"https:\/\/tuturpedia.com\/rumusmatematika\/volume-benda-putar\/","title":{"rendered":"Cara Menghitung Volume Benda Putar"},"content":{"rendered":"

Menghitung volume benda putar<\/a> melibatkan penggunaan rumus dan konsep integral, terutama ketika objek yang dihasilkan adalah hasil dari memutar suatu fungsi terhadap sumbu tertentu. Berikut adalah langkah-langkah umum untuk menghitung volume benda putar:<\/h2>\n

1. Pahami Jenis Benda Putar dan Fungsi yang Terlibat<\/a><\/strong><\/h3>\n

Identifikasi jenis benda putar yang ingin dihitung volumenya. Objek ini bisa berupa silinder, kerucut, atau benda putar lainnya. Pastikan juga untuk memahami fungsi matematika yang menggambarkan bentuk benda putar tersebut.<\/p>\n

\"%rumus<\/p>\n

2. Pilih Sumbu Rotasi<\/strong><\/h3>\n

Tentukan sumbu rotasi. Sumbu rotasi adalah sumbu di sepanjang mana objek diputar. Misalnya, untuk silinder, sumbu rotasi bisa sejajar dengan alasnya.<\/p>\n

3. Tentukan Batas-batas Integrasi<\/strong><\/h3>\n

Identifikasi batas-batas integrasi. Batas-batas ini adalah rentang nilai dari variabel independen (misalnya, \ufffd<\/span>x<\/span><\/span><\/span><\/span><\/span> atau \ufffd<\/span>y<\/span><\/span><\/span><\/span><\/span>) yang akan digunakan dalam integral.<\/p>\n

4. Tentukan Fungsi Luas Penampang<\/strong><\/h3>\n

Hitung fungsi luas penampang (\ufffd(\ufffd)<\/span>A<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span> atau \ufffd(\ufffd)<\/span>A<\/span>(<\/span>y<\/span>)<\/span><\/span><\/span><\/span><\/span>) pada setiap titik di sepanjang sumbu rotasi. Fungsi ini menggambarkan luas penampang potongan melintang benda putar.<\/p>\n

5. Hitung Integral dari Fungsi Luas Penampang<\/strong><\/h3>\n

Integral dari fungsi luas penampang ini memberikan volume benda putar. Jika sumbu rotasi adalah \ufffd<\/span>x<\/span><\/span><\/span><\/span><\/span>, rumus umumnya adalah:<\/p>\n

\ufffd=\ufffd\u222b\ufffd\ufffd[\ufffd(\ufffd)]2\u2009\ufffd\ufffd<\/span>V<\/span>=<\/span><\/span>\u03c0<\/span>\u222b<\/span>a<\/span><\/span><\/span>b<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>[<\/span>A<\/span>(<\/span>x<\/span>)<\/span>]2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>x<\/span><\/span><\/span><\/span><\/span><\/p>\n

Jika sumbu rotasi adalah \ufffd<\/span>y<\/span><\/span><\/span><\/span><\/span>, rumusnya menjadi:<\/p>\n

\ufffd=\ufffd\u222b\ufffd\ufffd[\ufffd(\ufffd)]2\u2009\ufffd\ufffd<\/span>V<\/span>=<\/span><\/span>\u03c0<\/span>\u222b<\/span>c<\/span><\/span><\/span>d<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>[<\/span>A<\/span>(<\/span>y<\/span>)<\/span>]2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>y<\/span><\/span><\/span><\/span><\/span><\/p>\n

Di sini, \ufffd<\/span>a<\/span><\/span><\/span><\/span><\/span>, \ufffd<\/span>b<\/span><\/span><\/span><\/span><\/span>, \ufffd<\/span>c<\/span><\/span><\/span><\/span><\/span>, dan \ufffd<\/span>d<\/span><\/span><\/span><\/span><\/span> adalah batas-batas integrasi.<\/p>\n

6. Hitung dan Evaluasi Integral<\/strong><\/h3>\n

Hitung integral menggunakan aturan integral yang sesuai. Jika integralnya kompleks, Anda mungkin perlu menggunakan teknik integral kalkulus atau alat perangkat lunak matematika.<\/p>\n

7. Gantilah Variabel dan Simpulkan Jawaban<\/strong><\/h3>\n

Gantilah variabel asli (seperti \ufffd<\/span>x<\/span><\/span><\/span><\/span><\/span> atau \ufffd<\/span>y<\/span><\/span><\/span><\/span><\/span>) dengan nilai-nilai batas-batasnya dan simpulkan hasilnya sebagai volume benda putar.<\/p>\n

8. Periksa Satuan Volume<\/strong><\/h3>\n

Pastikan untuk memeriksa satuan volume, karena volume dihitung dalam unit kubik (misalnya, cm\u00b3 atau m\u00b3).<\/p>\n

Contoh Perhitungan untuk Silinder<\/h3>\n

Misalkan kita ingin menghitung volume silinder dengan jari-jari \ufffd<\/span>r<\/span><\/span><\/span><\/span><\/span> dan tinggi \u210e<\/span>h<\/span><\/span><\/span><\/span><\/span>, dengan sumbu rotasi sejajar dengan alasnya.<\/p>\n

    \n
  1. Fungsi Luas Penampang:<\/strong> \ufffd(\ufffd)=\ufffd\ufffd2<\/span>A<\/span>(<\/span>x<\/span>)<\/span>=<\/span><\/span>\u03c0<\/span>r<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/li>\n
  2. Integral untuk Volume:<\/strong> \ufffd=\ufffd\u222b0\u210e\ufffd\ufffd2\u2009\ufffd\ufffd<\/span>V<\/span>=<\/span><\/span>\u03c0<\/span>\u222b<\/span>0<\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>r<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>x<\/span><\/span><\/span><\/span><\/span><\/li>\n
  3. Evaluasi Integral:<\/strong> \ufffd=\ufffd[13\ufffd\ufffd2\u210e]<\/span>V<\/span>=<\/span><\/span>\u03c0<\/span>[<\/span><\/span>3<\/span><\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>r<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/li>\n
  4. Simpulkan Jawaban:<\/strong> Volume silinder adalah 13\ufffd\ufffd2\u210e<\/span>3<\/span><\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>r<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span><\/span>.<\/li>\n<\/ol>\n

    Menggunakan langkah-langkah ini, Anda dapat menghitung volume benda putar dengan berbagai bentuk dan menggunakan integral untuk menangani bentuk yang lebih kompleks.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Menghitung volume benda putar melibatkan penggunaan rumus dan konsep integral, terutama ketika objek yang dihasilkan adalah hasil dari memutar suatu […]<\/p>\n","protected":false},"author":1,"featured_media":215,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[853,53,241,238,849,857,854,858,239,240,850,860,855,242,851,856,859,852],"class_list":["post-214","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-rumus-matematika","tag-keliling-segitiga","tag-keyword","tag-luas-persegi","tag-luas-persegi-panjang","tag-luas-segitiga","tag-luas-segitiga-siku-siku","tag-rumus-keliling-segitiga","tag-rumus-luas","tag-rumus-luas-persegi","tag-rumus-luas-persegi-panjang","tag-rumus-luas-segitiga","tag-rumus-luas-segitiga-siku-siku","tag-rumus-persegi","tag-rumus-persegi-panjang","tag-rumus-segitiga","tag-rumus-segitiga-siku-siku","tag-segitiga-sembarang","tag-segitiga-siku-siku"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/tuturpedia.com\/rumusmatematika\/wp-json\/wp\/v2\/posts\/214","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/tuturpedia.com\/rumusmatematika\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/tuturpedia.com\/rumusmatematika\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/tuturpedia.com\/rumusmatematika\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/tuturpedia.com\/rumusmatematika\/wp-json\/wp\/v2\/comments?post=214"}],"version-history":[{"count":4,"href":"https:\/\/tuturpedia.com\/rumusmatematika\/wp-json\/wp\/v2\/posts\/214\/revisions"}],"predecessor-version":[{"id":707,"href":"https:\/\/tuturpedia.com\/rumusmatematika\/wp-json\/wp\/v2\/posts\/214\/revisions\/707"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/tuturpedia.com\/rumusmatematika\/wp-json\/wp\/v2\/media\/215"}],"wp:attachment":[{"href":"https:\/\/tuturpedia.com\/rumusmatematika\/wp-json\/wp\/v2\/media?parent=214"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/tuturpedia.com\/rumusmatematika\/wp-json\/wp\/v2\/categories?post=214"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/tuturpedia.com\/rumusmatematika\/wp-json\/wp\/v2\/tags?post=214"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}