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döndürme bulunmaktadır.","Eksene teması varken döndürme ve iki grafik arasında kalan alanın x veya y ekseni etrafında döndürülmesi vardır.","İki grafik arasında kalan alanın x veya y ekseni etrafında döndürülmesine washer metodu denir, ancak bu disk metodunun bir alt kategorisidir."]},"beginTime":77,"endTime":238,"href":"/video/preview/411792444138273810?parent-reqid=1773545728742881-12075734129167029858-balancer-l7leveler-kubr-yp-vla-191-BAL&text=Datch+Integral&t=77&ask_summarization=1"},{"index":2,"title":"Doğrular Etrafında Döndürme","list":{"type":"unordered","items":["En zorlu kısım, x ve y ekseni etrafında döndürme dışında doğrular etrafında döndürmedir.","Disk metodunda x=a veya y=b doğruları etrafında döndürme de bulunmaktadır.","Shell metodunda da x ekseni, y ekseni ve doğrular etrafında döndürme çeşitleri bulunmaktadır."]},"beginTime":238,"endTime":348,"href":"/video/preview/411792444138273810?parent-reqid=1773545728742881-12075734129167029858-balancer-l7leveler-kubr-yp-vla-191-BAL&text=Datch+Integral&t=238&ask_summarization=1"},{"index":3,"title":"Hacim Hesaplama Şablonu","list":{"type":"unordered","items":["İntegralde hacim hesaplarken disk metodu ve shell metodu iki temel yöntemdir.","Disk metodunun altında x ekseni etrafında döndürme, y ekseni etrafında döndürme, washer metodu ve doğrular etrafında döndürme bulunmaktadır.","Shell metodunda da x ekseni, y ekseni ve doğrular etrafında döndürme çeşitleri incelenmektedir."]},"beginTime":348,"endTime":398,"href":"/video/preview/411792444138273810?parent-reqid=1773545728742881-12075734129167029858-balancer-l7leveler-kubr-yp-vla-191-BAL&text=Datch+Integral&t=348&ask_summarization=1"}],"linkTemplate":"/video/preview/411792444138273810?parent-reqid=1773545728742881-12075734129167029858-balancer-l7leveler-kubr-yp-vla-191-BAL&text=Datch+Integral&t=%%timestamp%%&ask_summarization=1"},"isAdultDoc":false,"relatedParams":{"text":"Calculus-I : İntegralde Hacim Hesaplama Çeşitleri (www.buders.com)","related_orig_text":"Datch Integral","related_porno":false,"related_less_3m_off":true,"client":"d2d","no_cnt":1,"related_src":"serp","related":"{\"porno\":false,\"vfp\":1,\"orig_text\":\"Datch 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üniversite matematiği derslerinden calculus-I dersine ait \" Çift Katlı İntegral Nedir ve Neyi Hesaplar? (Double Integral) \" videosudur. Hazırlayan: Kemal Duran (Matematik Öğretmeni)...","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/3009488/d78706d1d3b142eecdeb98d896edc06c/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/shXExwAAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"1","reqid":"1773545728742881-12075734129167029858-balancer-l7leveler-kubr-yp-vla-191-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1773545728742881-12075734129167029858-balancer-l7leveler-kubr-yp-vla-191-BAL&text=videoid:4738465065629411253","teaser":[{"list":{"type":"unordered","items":["Bu video, matematik eğitimi formatında çift katlı integral konusunu anlatan bir ders anlatımıdır.","Video, çift katlı integralin ne olduğunu açıklayarak başlıyor ve tek katlı integrallerden farkını vurguluyor. Çift katlı integralin üst üste iki defa integral alma işlemi olduğu, çok değişkenli fonksiyonların iki farklı değişkene göre integralini aldığı anlatılıyor. Daha sonra çift katlı integralin neyi hesapladığı konusuna geçiliyor: içine sadece 1 yazılırsa alan hesaplaması, içine çok değişkenli bir fonksiyon yazılırsa hacim hesaplaması yapıldığı açıklanıyor. Video, örnek integrallerle konuyu pekiştiriyor."]},"endTime":382,"title":"Çift Katlı İntegral Nedir ve Neyi Hesaplar?","beginTime":0}],"fullResult":[{"index":0,"title":"Çift Katlı İntegralin Tanımı","list":{"type":"unordered","items":["Çift katlı integral (double integral) üst üste iki defa integral alma işlemidir.","Tek katlı integrallerde bir değişkene bağlı integral alınırken, çift katlı integrallerde iki farklı değişkene (x ve y) göre integral alınır.","Çift katlı integral, çok değişkenli fonksiyonların (x, y gibi iki değişkene bağlı fonksiyonların) integralinin alınmasında kullanılır."]},"beginTime":1,"endTime":143,"href":"/video/preview/4738465065629411253?parent-reqid=1773545728742881-12075734129167029858-balancer-l7leveler-kubr-yp-vla-191-BAL&text=Datch+Integral&t=1&ask_summarization=1"},{"index":1,"title":"Çift Katlı İntegralin Hesapladığı Değerler","list":{"type":"unordered","items":["Tek katlı integraller alan hesaplamada kullanılırken, çift katlı integraller hem alan hem de hacim hesaplamada kullanılır.","Çift katlı integralin içerisinde \"1\" yazılırsa (veya hiçbir şey yazılmazsa), alan hesaplar.","Çift katlı integralin içerisinde çok değişkenli bir fonksiyon (f(x,y)) yazılırsa, hacim hesaplar."]},"beginTime":143,"endTime":307,"href":"/video/preview/4738465065629411253?parent-reqid=1773545728742881-12075734129167029858-balancer-l7leveler-kubr-yp-vla-191-BAL&text=Datch+Integral&t=143&ask_summarization=1"},{"index":2,"title":"Örnekler","list":{"type":"unordered","items":["İçerisinde hiçbir şey yazılmayan veya sadece \"1\" yazan çift katlı integral alan hesaplar.","İçerisinde çok değişkenli bir fonksiyon (örneğin x+2) yazan çift katlı integral hacim hesaplar.","Bu videoda çift katlı integralin ne olduğu ve ne hesapladığı anlatılırken, nasıl hesaplandığı ilerideki videolarda işlenecektir."]},"beginTime":307,"endTime":375,"href":"/video/preview/4738465065629411253?parent-reqid=1773545728742881-12075734129167029858-balancer-l7leveler-kubr-yp-vla-191-BAL&text=Datch+Integral&t=307&ask_summarization=1"}],"linkTemplate":"/video/preview/4738465065629411253?parent-reqid=1773545728742881-12075734129167029858-balancer-l7leveler-kubr-yp-vla-191-BAL&text=Datch+Integral&t=%%timestamp%%&ask_summarization=1"},"isAdultDoc":false,"relatedParams":{"text":"Calculus-II : Çift Katlı İntegral Nedir ve Neyi Hesaplar? 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üniversite matematiği derslerinden calculus-II dersine ait \" Yay Uzunluğuna Göre Çizgi İntegrali Hesaplama\" videosudur. Hazırlayan: Kemal Duran (Matematik Öğretmeni)...","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/466206/bada03e826a2946215c69c560bde6d03/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/pVXpGwEAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"2","reqid":"1773545728742881-12075734129167029858-balancer-l7leveler-kubr-yp-vla-191-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1773545728742881-12075734129167029858-balancer-l7leveler-kubr-yp-vla-191-BAL&text=videoid:12937631634814999737","teaser":[{"list":{"type":"unordered","items":["Bu video, bir eğitmen tarafından sunulan matematik dersi formatındadır. Eğitmen, yay uzunluğuna göre çizgi integrallerinin nasıl hesaplanacağını anlatmaktadır.","Videoda yay uzunluğuna göre çizgi integralinin formülü tanıtılmakta ve bu formülün uygulanması adım adım gösterilmektedir. Eğitmen, eğrinin parametrize edilmesi, ds değerinin hesaplanması ve integral sınırlarının belirlenmesi gibi temel adımları açıklamakta, iki boyutlu doğru parçası ve üç boyutlu eğri örnekleri üzerinden çözüm yapmaktadır.","Eğitmen, çemberin sağ yarısında saat yönü tersinde alınan bir çizgi integrali örneği üzerinden detaylı çözüm göstermekte ve bu konunun can damarı olduğunu belirterek, parametreze etme konusunda ayrı bir video çektiğini ifade etmektedir. Gelecek videolarda yay uzunluğuna göre çizgi integrali hesaplama örneklerinin devam edeceği belirtilmektedir."]},"endTime":1573,"title":"Yay Uzunluğuna Göre Çizgi İntegrali Hesaplama Dersi","beginTime":0}],"fullResult":[{"index":0,"title":"Yay Uzunluğuna Göre Çizgi İntegrali Formülü","list":{"type":"unordered","items":["Yay uzunluğuna göre çizgi integrali, f(x,y) veya f(x,y,z) fonksiyonunun bir eğri boyunca yay uzunluğuna göre integralini hesaplamayı ifade eder.","Formülde ds, yay uzunluğuna göre çizgi integrali alındığını gösteren kritik noktadır.","İki boyutlu ve üç boyutlu çizgi integral formülleri aynıdır, sadece ds yerine ds veya ds yazılır."]},"beginTime":1,"endTime":105,"href":"/video/preview/12937631634814999737?parent-reqid=1773545728742881-12075734129167029858-balancer-l7leveler-kubr-yp-vla-191-BAL&text=Datch+Integral&t=1&ask_summarization=1"},{"index":1,"title":"Parametrize Etme İşlemi","list":{"type":"unordered","items":["Yay uzunluğuna göre çizgi integrali hesaplamak için ilk adım eğrinin parametrize edilmesidir.","Parametrize etme sonucunda x ve y değerleri t değişkenine bağlı hale getirilir ve t'nin değişim aralığı (a'dan b'ye) belirlenir.","Parametrize etme sonucunda integral, t değişkenine bağlı bir integrale dönüştürülür."]},"beginTime":105,"endTime":315,"href":"/video/preview/12937631634814999737?parent-reqid=1773545728742881-12075734129167029858-balancer-l7leveler-kubr-yp-vla-191-BAL&text=Datch+Integral&t=105&ask_summarization=1"},{"index":2,"title":"ds Hesaplama","list":{"type":"unordered","items":["İkinci adımda sürekli ds ayarlanmalıdır.","ds, kök içinde (dx/dt)² + (dy/dt)² dt formülüyle hesaplanır.","Üç boyutlu durumda ds, kök içinde (dx/dt)² + (dy/dt)² + (dz/dt)² dt formülüyle hesaplanır."]},"beginTime":315,"endTime":492,"href":"/video/preview/12937631634814999737?parent-reqid=1773545728742881-12075734129167029858-balancer-l7leveler-kubr-yp-vla-191-BAL&text=Datch+Integral&t=315&ask_summarization=1"},{"index":3,"title":"Örnek Soru Çözümü","list":{"type":"unordered","items":["İlk örnek: ∫₋₂¹ 4(3t-2)³ ds integralini hesaplamak için önce doğru parçası parametrize edilir.","Doğru parçası için x = 3t-2 ve y = 3t-1 parametrize edilir, t'nin değişim aralığı 0'dan 1'e kadardır.","İntegral hesaplanarak sonuç 12√2(3-15/42) olarak bulunur."]},"beginTime":492,"endTime":992,"href":"/video/preview/12937631634814999737?parent-reqid=1773545728742881-12075734129167029858-balancer-l7leveler-kubr-yp-vla-191-BAL&text=Datch+Integral&t=492&ask_summarization=1"},{"index":4,"title":"Yay Uzunluğuna Göre Çizgi İntegrali Hesaplama","list":{"type":"unordered","items":["Çemberin sağ yarısında saat yönü tersi alınan bir yay uzunluğuna göre çizgi integrali hesaplanmaktadır.","Çember ve elips sorulduğunda mutlaka yön belirtilmesi gerekir.","ds yazıldığında yay uzunluğuna göre çizgi integral hesaplandığı anlaşılır."]},"beginTime":998,"endTime":1072,"href":"/video/preview/12937631634814999737?parent-reqid=1773545728742881-12075734129167029858-balancer-l7leveler-kubr-yp-vla-191-BAL&text=Datch+Integral&t=998&ask_summarization=1"},{"index":5,"title":"Çemberin Parametrize Edilmesi","list":{"type":"unordered","items":["Çemberin parametreze edilme kuralı x = r cos(t) ve y = r sin(t) şeklindedir.","Saat yönü tersinde dönüyorsa y'nin önüne eksi gelir.","Çemberin yarıçapı 4 olduğundan x = 4 cos(t) ve y = 4 sin(t) olarak parametrize edilir."]},"beginTime":1072,"endTime":1188,"href":"/video/preview/12937631634814999737?parent-reqid=1773545728742881-12075734129167029858-balancer-l7leveler-kubr-yp-vla-191-BAL&text=Datch+Integral&t=1072&ask_summarization=1"},{"index":6,"title":"İntegralin Hesaplanması","list":{"type":"unordered","items":["İntegralin sınırları t = -π/2'den t = π/2'ye kadardır.","ds = √((x'in t'ye göre türevinin karesi + y'nin t'ye göre türevinin karesi) dt formülü kullanılarak ds = 4 dt olarak bulunur.","İntegral 4⁶∫₋π/²² cos(2t) sin⁴(4sin(t)) dt şeklinde yazılır ve değişken değiştirme yöntemiyle çözülür."]},"beginTime":1188,"endTime":1470,"href":"/video/preview/12937631634814999737?parent-reqid=1773545728742881-12075734129167029858-balancer-l7leveler-kubr-yp-vla-191-BAL&text=Datch+Integral&t=1188&ask_summarization=1"},{"index":7,"title":"Çizgi İntegrali Hesaplama Adımları","list":{"type":"unordered","items":["Yay uzunluğuna göre çizgi integrali hesaplamada öncelikle eğri parametrize edilir.","T'nin aralığı ve x, y yerine gelecek t'li ifadeler belirlendikten sonra integral fonksiyonuna yerleştirilir.","ds yerine kullanılan formül kullanılarak t'ye bağlı bir integral elde edilir ve bu integral hesaplanarak f(x,y) fonksiyonunun yay uzunluğuna göre integrali 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