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Öğretmen, belirli integral konusunu detaylı bir şekilde anlatmaktadır.","Video, belirli integralin tanımı, hesaplanma yöntemleri ve özellikleri üzerine odaklanmaktadır. İçerikte parabol ve doğru gibi eğrilerin x eksenine sınırladığı alanların hesaplanması, integral alma kuralları, alan hesaplamaları ve değişken değiştirme teknikleri örneklerle açıklanmaktadır. Ayrıca polinomlarla ilgili problemler ve türev-integral ilişkisi de ele alınmaktadır.","Video, KPSS sınavına hazırlık amacıyla hazırlanmış olup, 138. sorudan 146. soruya kadar olan bir dizi problemi içermektedir. 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çıkarılır."]},"beginTime":156,"endTime":322,"href":"/video/preview/8544476991605607823?parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL&text=Datch+Integral&t=156&ask_summarization=1"},{"index":3,"title":"Örneklerle Belirli İntegral","list":{"type":"unordered","items":["Örnek olarak y=3x² parabolünün 1'den 3'e kadar sınırladığı bölgenin alanı hesaplanıyor.","İntegral hesaplandıktan sonra üst sınır (3) yerine yazıp sonuç 27, alt sınır (1) yerine yazıp sonuç 1 bulunuyor.","İkinci örnek olarak y=2x+3 doğrusunun 1'den 3'e kadar sınırladığı bölgenin alanı hesaplanıyor ve sonuç 14 birim kare bulunuyor."]},"beginTime":322,"endTime":421,"href":"/video/preview/8544476991605607823?parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL&text=Datch+Integral&t=322&ask_summarization=1"},{"index":4,"title":"Belirli İntegral Hesaplama","list":{"type":"unordered","items":["Belirli integral hesaplaması için önce integrali alıp, sonra üst sınır değerini yazıp eksi işaretini koyarak alt sınır değerini yazmak gerekir.","İntegral hesaplaması temel olarak iki değer yazıp çıkarmaktır.","İntegral alma işlemi kolay, ancak belirli integralde üst ve alt sınır değerlerini doğru hesaplamak önemlidir."]},"beginTime":424,"endTime":487,"href":"/video/preview/8544476991605607823?parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL&text=Datch+Integral&t=424&ask_summarization=1"},{"index":5,"title":"İntegral Örneği","list":{"type":"unordered","items":["İntegral hesaplaması için önce fonksiyonun integrali alınır, sonra üst sınır değeri yerine yazılır ve eksi işaretiyle alt sınır değeri çıkarılır.","İntegral hesaplaması sonucunda elde edilen değer, belirli bir aralıkta fonksiyonun altında kalan alanı temsil eder.","İntegral hesaplaması, belirli bir aralıkta fonksiyonun altında kalan alanı hesaplamak için kullanılır."]},"beginTime":487,"endTime":632,"href":"/video/preview/8544476991605607823?parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL&text=Datch+Integral&t=487&ask_summarization=1"},{"index":6,"title":"İntegral Alma Kuralları","list":{"type":"unordered","items":["İntegral alma kuralları, belirli integral hesaplamalarında kullanılır.","İntegral alma işlemi, fonksiyonun üst sınırını bir artırıp, yeni üsse bölerek yapılır.","İntegral alma işlemi, belirli integral hesaplamalarında kolay bir yöntemdir."]},"beginTime":632,"endTime":698,"href":"/video/preview/8544476991605607823?parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL&text=Datch+Integral&t=632&ask_summarization=1"},{"index":7,"title":"İntegral Örnekleri","list":{"type":"unordered","items":["İntegral hesaplamalarında, fonksiyonun üst sınırını bir artırıp, yeni üsse bölerek hesaplama yapılır.","İntegral hesaplamalarında, paydaları eşitleyerek ve toplama çıkarma işlemleri yaparak sonuç bulunabilir.","İntegral hesaplamalarında, fonksiyonun üst sınırını bir artırıp, yeni üsse bölerek hesaplama yapılır."]},"beginTime":698,"endTime":1049,"href":"/video/preview/8544476991605607823?parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL&text=Datch+Integral&t=698&ask_summarization=1"},{"index":8,"title":"İntegral Problemleri","list":{"type":"unordered","items":["Fonksiyonun grafiği verilmiş ve belirli integral hesaplanıyor: ∫₀⁴ f'(x) dx = f(4) - f(0) = 3 - (-3) = 6.","Karmakarışık bir integral problemi çözülüyor: ∫₋₂³ (f(x)/x)' dx = ∫₋₂³ f(x)/2 dx = f(3)/2 - f(-2)/2 = 3/2 - (-4)/2 = -4/3.","Belirli integral alma kuralı kullanılarak ∫ₐᵇ (4x² - 2x) dx = 2x³ - 2x² = 2b² - 2a² = -8 denklemi çözülüyor ve b-a = 4 bilgisi kullanılarak b = 2, a = -2 bulunuyor."]},"beginTime":1054,"endTime":1384,"href":"/video/preview/8544476991605607823?parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL&text=Datch+Integral&t=1054&ask_summarization=1"},{"index":9,"title":"İntegral Özellikleri","list":{"type":"unordered","items":["İntegral alma kuralı hatırlatılıyor: ∫₂⁶ 6x dx = 6x² = 36 - 12 = 24.","Belirli integralin sonucu sabit bir sayı olduğunda, bu sabit sayının türevi sıfırdır.","Parantez içindeki integral hesaplanıyor: ∫₁³ (x dy) = 2y = 3x - 2y = 2x."]},"beginTime":1384,"endTime":1579,"href":"/video/preview/8544476991605607823?parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL&text=Datch+Integral&t=1384&ask_summarization=1"},{"index":10,"title":"Türev ve İntegral Uygulaması","list":{"type":"unordered","items":["f''(x) fonksiyonunun integrali hesaplanıyor: ∫₋₂³ f''(x) dx = f'(3) - f'(-2).","f'(3) = 0, f'(-2) = 4/6 = 1/3 olarak bulunuyor ve sonuç -1/3 olarak hesaplanıyor.","Polinom P(x) için P(x) = x - 1 ile bölümünden kalan hesaplanıyor."]},"beginTime":1579,"endTime":1691,"href":"/video/preview/8544476991605607823?parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL&text=Datch+Integral&t=1579&ask_summarization=1"},{"index":11,"title":"İntegral ve Polinom Problemi","list":{"type":"unordered","items":["İntegral alınarak P(x) polinomu bulunuyor ve P(x) = 2x² - 3x + n şeklinde ifade ediliyor.","P(0) = n ve P(-1) = 5 + n değerleri hesaplanarak n = -6 bulunuyor.","P(1) değeri hesaplanarak -7 olarak bulunuyor ve sorunun doğru cevabı E şıkkı olarak belirleniyor."]},"beginTime":1695,"endTime":1869,"href":"/video/preview/8544476991605607823?parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL&text=Datch+Integral&t=1695&ask_summarization=1"},{"index":12,"title":"İkinci İntegral Problemi","list":{"type":"unordered","items":["Belirli integral kullanılarak P(x) polinomu tekrar bulunuyor ve P(x) = 2x² - 3x + n şeklinde ifade ediliyor.","P(0) = n ve P(-1) = 5 + n değerleri hesaplanarak n = -6 bulunuyor.","P(1) değeri hesaplanarak -7 olarak bulunuyor ve sorunun doğru cevabı E şıkkı olarak belirleniyor."]},"beginTime":1869,"endTime":1982,"href":"/video/preview/8544476991605607823?parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL&text=Datch+Integral&t=1869&ask_summarization=1"},{"index":13,"title":"Fonksiyonun Türevi ve Minimum Noktası","list":{"type":"unordered","items":["f(x) fonksiyonunun türevi bulunarak yerel minimum noktası hesaplanıyor.","İntegral alınarak f(x) = 6x² + 6x - 3 olarak bulunuyor ve f'(x) = 12x + 6 olarak hesaplanıyor.","f'(x) = 0 denkleminden x = -1/2 bulunuyor ve bu değer yerel minimum noktası olarak belirleniyor."]},"beginTime":1982,"endTime":2147,"href":"/video/preview/8544476991605607823?parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL&text=Datch+Integral&t=1982&ask_summarization=1"},{"index":14,"title":"İntegralin Özellikleri","list":{"type":"unordered","items":["Belirli integralin çok önemli ve değerli özellikleri var.","Özellikle alan hesaplamalarında çok kullanılacak özellikleri bulunmaktadır.","İntegralin özellikleri konusu daha sonra ele alınacak."]},"beginTime":2147,"endTime":2167,"href":"/video/preview/8544476991605607823?parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL&text=Datch+Integral&t=2147&ask_summarization=1"}],"linkTemplate":"/video/preview/8544476991605607823?parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL&text=Datch+Integral&t=%%timestamp%%&ask_summarization=1"},"isAdultDoc":false,"relatedParams":{"text":"İntegral 12 Belirli İntegral | 80 Günde 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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/2808942/bada03e826a2946215c69c560bde6d03/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/pVXpGwEAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"6","reqid":"1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-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=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-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=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-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=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-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=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-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=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-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=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-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=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-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 bulunur."]},"beginTime":1470,"endTime":1568,"href":"/video/preview/12937631634814999737?parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL&text=Datch+Integral&t=1470&ask_summarization=1"}],"linkTemplate":"/video/preview/12937631634814999737?parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL&text=Datch+Integral&t=%%timestamp%%&ask_summarization=1"},"isAdultDoc":false,"relatedParams":{"text":"Calculus-II : Yay Uzunluğuna Göre Çizgi İntegrali Hesaplama","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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Eğitmen önce integral alma bölgesini çizerek sınırları belirleyip, ardından dairesel alan olduğu için kutupsal koordinatları kullanarak integral sınırlarını ve integral alma sırasını açıklamaktadır. Son olarak, integral hesaplaması yapılarak sonucun π/8 sin(1) olduğu gösterilmektedir."]},"endTime":591,"title":"Çift Katlı İntegral Örnek Soru Çözümü","beginTime":0}],"fullResult":[{"index":0,"title":"Çift Katlı İntegral Sorusu","list":{"type":"unordered","items":["Video, çift katlı integral konusunda bir örnek soruyu inceleyecektir.","Soruda, birinci bölgede y=x doğrusu, x ekseni ve y=√(1-x²) eğrisi ile sınırlı bir bölgede cos(x²+y²) integralinin sonucu bulunması istenmektedir.","Çift katlı integral sorularında ilk adım, integral alma bölgesini çizmek ve bu bölge sayesinde integralin sınırlarını ve sırasını belirlemektir."]},"beginTime":1,"endTime":43,"href":"/video/preview/9426788986556539746?parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL&text=Datch+Integral&t=1&ask_summarization=1"},{"index":1,"title":"İntegral Alma Bölgesinin Çizimi","list":{"type":"unordered","items":["Birinci bölgede, x-y düzleminin birinci bölgesi kastedilmektedir.","İntegral alma bölgesi, y=x doğrusu, x ekseni ve y=√(1-x²) eğrisi ile sınırlıdır.","Çember y=√(1-x²) denklemi, yarıçapı 1 olan bir yarım çemberdir ve x ekseninin üstünde kalır."]},"beginTime":43,"endTime":203,"href":"/video/preview/9426788986556539746?parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL&text=Datch+Integral&t=43&ask_summarization=1"},{"index":2,"title":"Kutupsal Koordinatların Kullanımı","list":{"type":"unordered","items":["İntegral alma bölgesi dairesel olduğu için kutupsal koordinatlar kullanılmalıdır.","Kutupsal koordinatlarda r yarıçap, 0 ile 1 arasında değişir.","θ açısı, x ekseni ile pozitif yönde yaptığı açıdır ve 0 ile π/4 arasında değişir."]},"beginTime":203,"endTime":341,"href":"/video/preview/9426788986556539746?parent-reqid=1769100448420020-118248045588830639-balancer-l7leveler-kubr-yp-vla-18-BAL&text=Datch+Integral&t=203&ask_summarization=1"},{"index":3,"title":"İntegralin 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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. 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