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Öğretmen, öğrencilere hitap ederek konuyu adım adım açıklamaktadır.","Videoda integralin alan hesaplama amacıyla kullanıldığı anlatılmaktadır. İçerik, teorik bilgilerle başlayıp, belirli integral ile alan hesaplama, eğrinin altında kalan alanların hesaplanması, iki fonksiyon arasındaki bölgenin alanının bulunması ve grafik çizimi gerektiren alan hesaplama örnekleri şeklinde ilerlemektedir.","Öğretmen, Reman adlı bir matematikçi tarafından geliştirilen dikdörtgenler yöntemiyle integralin nasıl geliştirildiğini açıklamakta ve her bir soruyu çözerken integral alma kurallarını, sınırların nasıl belirleneceğini ve sonuçların nasıl yorumlanacağını detaylı şekilde göstermektedir. Dersin sonunda, bir sonraki derste hacim hesaplaması yapılacağı belirtilmektedir."]},"endTime":1212,"title":"İntegralin Alan Hesaplama Amacı Dersi","beginTime":0}],"fullResult":[{"index":0,"title":"İntegralin Amacı","list":{"type":"unordered","items":["İntegralin asıl amacı alan hesaplamaktır.","İntegral, verilen fonksiyonun grafiğinin altında kalan bölgenin alanını bulmaya yarar.","Belirli integral, fonksiyonun grafiğinin altında kalan alanı hesaplamak için AB aralığında integral alınarak bulunur."]},"beginTime":4,"endTime":58,"href":"/video/preview/7727012358414870190?parent-reqid=1769166208131880-13764100407186189801-balancer-l7leveler-kubr-yp-sas-175-BAL&text=Integral_RoA&t=4&ask_summarization=1"},{"index":1,"title":"İntegralin Tarihi","list":{"type":"unordered","items":["Reman adlı bir matematikçi, eğrinin altına ve üstüne dikdörtgenler çizerek integral metodunu keşfetmiştir.","Reman, alt ve üstteki dikdörtgenleri birbirine yaklaştırdı ve integral metodunu buldu.","Sonuç olarak, fonksiyonun bir kuvvetinin integrali, o fonksiyonun grafiğinin altındaki alanı verir."]},"beginTime":58,"endTime":102,"href":"/video/preview/7727012358414870190?parent-reqid=1769166208131880-13764100407186189801-balancer-l7leveler-kubr-yp-sas-175-BAL&text=Integral_RoA&t=58&ask_summarization=1"},{"index":2,"title":"İntegral ile Alan Hesaplama","list":{"type":"unordered","items":["İntegral kullanarak belirli bir aralıkta (x'den 3'e kadar) fonksiyonun (x²+1) altında kalan alan hesaplanabilir.","İntegral hesaplandıktan sonra sınırlar yerine konularak alan değeri bulunur, bu örnekte alan 12 birim kare olarak hesaplanmıştır.","Eğer fonksiyon x ekseninin altındaysa, bulunan integral değeri önüne eksi işareti konularak pozitif bir alan değeri elde edilir."]},"beginTime":108,"endTime":365,"href":"/video/preview/7727012358414870190?parent-reqid=1769166208131880-13764100407186189801-balancer-l7leveler-kubr-yp-sas-175-BAL&text=Integral_RoA&t=108&ask_summarization=1"},{"index":3,"title":"İki Fonksiyon Arasındaki Alan Hesaplama","list":{"type":"unordered","items":["İki farklı fonksiyonun grafiği arasında kalan alan hesaplanırken, üstteki fonksiyondan alttaki fonksiyon çıkarılıp integral alınır.","İntegral hesaplaması için önce fonksiyonların sınırları belirlenir, sonra üstteki fonksiyondan alttaki fonksiyon çıkarılıp integral alınır.","İntegral hesaplandıktan sonra sınırlar yerine konularak sayısal sonuç bulunur, bu örnekte alan 36 birim kare olarak hesaplanmıştır."]},"beginTime":365,"endTime":738,"href":"/video/preview/7727012358414870190?parent-reqid=1769166208131880-13764100407186189801-balancer-l7leveler-kubr-yp-sas-175-BAL&text=Integral_RoA&t=365&ask_summarization=1"},{"index":4,"title":"Karmaşık Alan Hesaplama","list":{"type":"unordered","items":["Grafiği çizilmeyen fonksiyonlar için önce grafik çizilerek alan hesaplaması yapılır.","İntegral hesaplaması için x'in alabileceği en küçük ve en büyük değerler belirlenir.","İntegral hesaplandıktan sonra sınırlar yerine konularak sayısal sonuç bulunur, bu örnekte alan 11/3 birim kare olarak hesaplanmıştır."]},"beginTime":738,"endTime":895,"href":"/video/preview/7727012358414870190?parent-reqid=1769166208131880-13764100407186189801-balancer-l7leveler-kubr-yp-sas-175-BAL&text=Integral_RoA&t=738&ask_summarization=1"},{"index":5,"title":"İntegral ile Alan Hesaplama","list":{"type":"unordered","items":["İki fonksiyonun grafiği çiziliyor: y = x² - 8 ve y = -x² + 6x.","İki eğri arasındaki alan hesaplanmak isteniyor.","Alan hesaplamak için üstteki fonksiyondan alttaki fonksiyon çıkarılıp integral alınması gerekiyor."]},"beginTime":897,"endTime":997,"href":"/video/preview/7727012358414870190?parent-reqid=1769166208131880-13764100407186189801-balancer-l7leveler-kubr-yp-sas-175-BAL&text=Integral_RoA&t=897&ask_summarization=1"},{"index":6,"title":"İntegral Sınırlarının Belirlenmesi","list":{"type":"unordered","items":["İntegral sınırları, iki fonksiyonun kesişim noktaları olarak bulunuyor.","Fonksiyonlar eşitlenerek x² - 8 = -x² + 6x denklemi çözülüyor.","Denklemin çözümü x = -1 ve x = 4 olarak bulunuyor, bu da integral sınırları oluyor."]},"beginTime":997,"endTime":1065,"href":"/video/preview/7727012358414870190?parent-reqid=1769166208131880-13764100407186189801-balancer-l7leveler-kubr-yp-sas-175-BAL&text=Integral_RoA&t=997&ask_summarization=1"},{"index":7,"title":"İntegral Hesaplama ve Sonuç","list":{"type":"unordered","items":["(-2x³/3 + 3x² + 8x) fonksiyonunun -1 ile 4 arasında integrali hesaplanıyor.","İntegral hesaplaması sonucunda bölgenin alanı 125/3 olarak bulunuyor.","İntegralin temel amacı alan ve hacim hesaplaması yapabilmek için öğrenildiği 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Öğretmen, beşinci derste kısmi integrasyonu ele alacağını belirtiyor.","Videoda kısmi integrasyonun ne zaman kullanılacağı, LAPT (Logaritma, Ark, Polinom, Trigonometri, Üstel fonksiyon) akrostijini kullanarak u ve dv'yi nasıl seçeceğimiz ve formülün nasıl uygulanacağı adım adım gösterilmektedir. Öğretmen, x çarpı kosinüs x dx, x çarpı artanjant x dx ve x çarpı ln x dx gibi örnekler üzerinden kısmi integrasyonun nasıl yapılacağını detaylı şekilde açıklamaktadır.","Videoda ayrıca logaritmik fonksiyonların integrali ve değişken değiştirme yöntemleri de ele alınmaktadır. Video, bir sonraki derste belirli integralin anlatılacağı bilgisiyle sonlanmaktadır."]},"endTime":905,"title":"Kısmi İntegrasyon Eğitim Videosu","beginTime":0}],"fullResult":[{"index":0,"title":"Kısmi İntegrasyon Kavramı","list":{"type":"unordered","items":["Kısmi integrasyon, basit olmayan veya değişken değiştirme yöntemiyle çözülemeyen integral problemlerinde kullanılır.","Kısmi integrasyonda, integralin bir parçası u olarak, kalan parçası dv olarak belirlenir ve formül u*v - ∫v*du şeklinde kullanılır.","u'nun seçilimi LAPT kuralına göre yapılır: Logaritma, Ark (sinüs, kosinüs), Polinom, Trigonometri ve Üstel fonksiyonlar bu sırayla tercih edilir."]},"beginTime":6,"endTime":108,"href":"/video/preview/18376548993141115334?parent-reqid=1769166208131880-13764100407186189801-balancer-l7leveler-kubr-yp-sas-175-BAL&text=Integral_RoA&t=6&ask_summarization=1"},{"index":1,"title":"Kısmi İntegrasyon Örnekleri","list":{"type":"unordered","items":["∫x*cos(x) dx integralinde, LAPT kuralına göre x (polinom) u olarak, cos(x) dx ise dv olarak seçilir.","∫tan⁻¹(x) dx integralinde, LAPT kuralına göre tan⁻¹(x) u olarak, dx ise dv olarak seçilir.","∫ln(x)/x dx integralinde, LAPT kuralına göre ln(x) u olarak, 1/x dx ise dv olarak seçilir."]},"beginTime":108,"endTime":552,"href":"/video/preview/18376548993141115334?parent-reqid=1769166208131880-13764100407186189801-balancer-l7leveler-kubr-yp-sas-175-BAL&text=Integral_RoA&t=108&ask_summarization=1"},{"index":2,"title":"Kısmi İntegrasyon Uygulamaları","list":{"type":"unordered","items":["Kısmi integrasyon sonucunda elde edilen integral daha basit hale gelmelidir, aksi takdirde tekrar kısmi integrasyon uygulanmalıdır.","Logaritmik fonksiyonlar kısmi integrasyonda LAPT kuralına göre tercih edilir ve genellikle elenmek için e tabanına dönüştürülür.","Kısmi integrasyon uygulamasında, u ve v'nin türevleri ve integrali doğru şekilde hesaplanmalı, ardından formülde yerine 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