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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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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 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kullanılır."]},"beginTime":487,"endTime":632,"href":"/video/preview/8544476991605607823?parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL&text=Datch+Integral&t=2147&ask_summarization=1"}],"linkTemplate":"/video/preview/8544476991605607823?parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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/222140/bada03e826a2946215c69c560bde6d03/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/pVXpGwEAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"6","reqid":"1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL&text=Datch+Integral&t=1470&ask_summarization=1"}],"linkTemplate":"/video/preview/12937631634814999737?parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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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Hazırlayan: Kemal Duran (Matematik Öğretmeni) http://www.buders.com/kadromuz.html...","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/222140/bada03e826a2946215c69c560bde6d03/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/_chU_wAAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"10","reqid":"1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL&text=videoid:9426788986556539746","teaser":[{"list":{"type":"unordered","items":["Bu video, bir matematik eğitim içeriğidir. Eğitmen, çift katlı integral konusunda örnek bir soru çözümü sunmaktadır.","Videoda, birinci bölgede y = x doğrusu, x ekseni ve y = √(1-x²) eğrisi ile sınırlı bir bölgede cos(x² + y²) integralinin nasıl hesaplanacağı adım adım anlatılmaktadır. 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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL&text=Datch+Integral&t=203&ask_summarization=1"},{"index":3,"title":"İntegralin 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üniversite matematiği derslerinden calculus-I dersine ait \"Çift Katlı İntegralde Değişken Değiştirme (Change of Variables in Double Integral) \" videosudur. Hazırlayan: Kemal Duran...","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/222140/bada03e826a2946215c69c560bde6d03/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/oZOO_wAAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"17","reqid":"1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL&text=videoid:7557418428767430028","teaser":[{"list":{"type":"unordered","items":["Bu video, bir eğitmen tarafından sunulan matematik dersi formatında olup, çift katlı integralde değişken değiştirme konusunu anlatmaktadır.","Videoda, çift katlı integralde değişken değiştirme yönteminin ne olduğu, neden ihtiyaç duyulduğu ve nasıl uygulanacağı adım adım açıklanmaktadır. Eğitmen, kartezyen koordinatlarda çözüm zorlu olan bölgelerde x ve y değişkenlerini u ve v gibi farklı değişkenlere dönüştürerek integralin daha kolay çözülebilmesi konusunu ele almaktadır. Video, R bölgesi için x = 2u ve y = 4 + 2u dönüşümü örneği üzerinden jakobiyan hesaplaması, bölge çizimi ve yeni sınırların belirlenmesi gibi temel adımları göstermektedir.","Videoda ayrıca değişken değiştirme formülleri, jakobiyan kavramı ve dönüşümlerin nasıl yapılacağı detaylı şekilde anlatılmakta, bir sonraki beş örnek çözüm videosunun geleceğini belirtmektedir."]},"endTime":2090,"title":"Çift Katlı İntegralde Değişken Değiştirme Dersi","beginTime":0}],"fullResult":[{"index":0,"title":"Çift Katlı İntegralde Değişken Değiştirme Kavramı","list":{"type":"unordered","items":["Çift katlı integralde değişken değiştirme işlemi, kartezyen koordinatlarda yazılan çift katlı integralleri farklı koordinat sistemlerine dönüştürme yöntemidir.","Kartezyen koordinatlarda hesaplama zorluğundan dolayı, x ve y değişkenleri u ve v gibi yeni değişkenlere dönüştürülür.","Kutupsal koordinatlara dönüşüm de bir değişken değiştirme yöntemidir, ancak özel kuralları olan bir yöntemdir."]},"beginTime":1,"endTime":141,"href":"/video/preview/7557418428767430028?parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL&text=Datch+Integral&t=1&ask_summarization=1"},{"index":1,"title":"Değişken Değiştirme Gerekliliği","list":{"type":"unordered","items":["Değişken değiştirme, x ve y'ye bağlı çift katlı integralin hesaplanmasında yaşanan zorlukları çözmek için yapılır.","İntegralin hesaplanmasında zorluk, R bölgesinin x ve y'nin sınırlarını belirlemede zorlu bir alan olmasıdır.","Değişken değiştirme, R bölgesini u ve v değişkenlerine ait daha basit ve sınırları kolayca belirlenebilir bir alana dönüştürür."]},"beginTime":141,"endTime":464,"href":"/video/preview/7557418428767430028?parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL&text=Datch+Integral&t=141&ask_summarization=1"},{"index":2,"title":"Değişken Değiştirme Formülü","list":{"type":"unordered","items":["Değişken değiştirme formülü: ∫∫_R f(x,y) dA = ∫∫_S f(g(u,v), h(u,v)) |J| du dv şeklindedir.","Jakobiyan (J), x ve y'nin u ve v'ye göre kısmi türevlerinden oluşan bir determinanttır: J = |∂x/∂u ∂x/∂v| = (∂x/∂u)(∂y/∂v) - (∂x/∂v)(∂y/∂u).","Jakobiyan negatif çıkarsa mutlak değer alınarak pozitif bir değer elde edilir."]},"beginTime":464,"endTime":863,"href":"/video/preview/7557418428767430028?parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL&text=Datch+Integral&t=464&ask_summarization=1"},{"index":3,"title":"Örnek Soru","list":{"type":"unordered","items":["Örnek soruda, R bölgesi x=2, y=x+4 ve y=x-4 doğrularıyla sınırlanan kapalı bölge olarak verilmiştir.","İntegral, x=2u, y=4+2v dönüşümü yaparak çözülmek istenmektedir."]},"beginTime":863,"endTime":915,"href":"/video/preview/7557418428767430028?parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL&text=Datch+Integral&t=863&ask_summarization=1"},{"index":4,"title":"Değişken Değiştirme Yöntemi","list":{"type":"unordered","items":["R bölgesi için xy'li integral, x=2u ve y=4v+2u dönüşümüyle çözülecek.","İlk adım jakobiyanın hesaplanmasıdır, ancak bu adım en sonda da yapılabilir.","Jakobiyan hesaplanırken x'in u'ya göre kısmi türevi 2, x'in v'ye göre kısmi türevi 0, y'nin u'ya göre kısmi türevi 4, y'nin v'ye göre kısmi türevi 4 olarak bulunur."]},"beginTime":952,"endTime":1067,"href":"/video/preview/7557418428767430028?parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL&text=Datch+Integral&t=952&ask_summarization=1"},{"index":5,"title":"Jakobiyan Hesaplama","list":{"type":"unordered","items":["Jakobiyan determinantı hesaplandığında 2×4=8 sonucu elde edilir.","Jakobiyan mutlak değer içinde kullanılır ve sonuç pozitif olduğu için |8|=8 olur.","Değişken değiştirme sorularında ilk adım jakobiyanı hesaplamak, ikinci adım x ve y'li bölgeyi çizmektir."]},"beginTime":1067,"endTime":1140,"href":"/video/preview/7557418428767430028?parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL&text=Datch+Integral&t=1067&ask_summarization=1"},{"index":6,"title":"R Bölgesinin Çizimi","list":{"type":"unordered","items":["R bölgesi x=0'dan x=2'ye, y=x'ten y=x+4'e kadar olan bölgedir.","y=x doğrusu orijinden geçen 45 derecelik açı yapan doğrudur.","y=x+4 doğrusu y eksenini 4'te, x eksenini -4'te keser ve y=x doğrusuna paraleldir."]},"beginTime":1140,"endTime":1309,"href":"/video/preview/7557418428767430028?parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL&text=Datch+Integral&t=1140&ask_summarization=1"},{"index":7,"title":"UV Düzlemine Dönüşüm","list":{"type":"unordered","items":["x=0 sınırı, x=2u dönüşümünde u=0 olarak elde edilir.","y=x sınırı, y=4v+2u ve x=2u dönüşümlerinde v=0 olarak bulunur.","y=x+4 sınırı, aynı dönüşümlerle v=1 olarak hesaplanır."]},"beginTime":1309,"endTime":1579,"href":"/video/preview/7557418428767430028?parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL&text=Datch+Integral&t=1309&ask_summarization=1"},{"index":8,"title":"UV Düzlemi ve İntegral Sınırları","list":{"type":"unordered","items":["UV düzleminde u 0'dan 1'e, v 0'dan 1'e kadar değişir ve kare şeklinde bir bölge oluşur.","İntegral sınırları belirlendikten sonra, integral R bölgesinden UV bölgesine dönüştürülür.","İntegralde x=2u, y=4v+2u dönüşümleri ve jakobiyan |8| kullanılarak son integral formu elde edilir."]},"beginTime":1579,"endTime":1816,"href":"/video/preview/7557418428767430028?parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL&text=Datch+Integral&t=1579&ask_summarization=1"},{"index":9,"title":"Çift Katlı İntegralde Değişken Değiştirme","list":{"type":"unordered","items":["İntegral hesaplaması için u'ya göre entegrasyon yapılarak sonuç bulunuyor.","İntegral hesaplaması sonucunda 83 olarak sonuç elde ediliyor.","Bu videodan sonra seri biçimde değişken değiştirme ile ilgili tahmini beş örnek çözülecek."]},"beginTime":1839,"endTime":1929,"href":"/video/preview/7557418428767430028?parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL&text=Datch+Integral&t=1839&ask_summarization=1"},{"index":10,"title":"Değişken Değiştirme Yönteminin Amacı","list":{"type":"unordered","items":["Çift katlı integralde değişken değiştirme, x-y bölgesindeki alanın sınırlarını belirlemede zorlu durumlarda kullanılır.","Değişken değiştirme yaparken x ve y'yi u ve v cinsinden ifade edip, içerdaki fonksiyona yerleştiririz.","İntegrali hesaplamak için jakobiyan hesaplanması gerekir."]},"beginTime":1929,"endTime":1997,"href":"/video/preview/7557418428767430028?parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL&text=Datch+Integral&t=1929&ask_summarization=1"},{"index":11,"title":"Değişken Değiştirme Yönteminin Uygulanması","list":{"type":"unordered","items":["Önce jakobiyan hesaplanır, sonra x ve y'ye ait bölge çizilir.","x ve y'lerin sınır fonksiyonlarına dönüşümler uygulanarak u ve v'ye ait sınırlar elde edilir.","Bu bölümde en sık yapılan hata ve en zorlu kısımdır.","Elde edilen u ve v sınırları kullanılarak yeni alan çizilir ve integral sınırları belirlenir.","İntegralde x ve y yerine u ve v'li ifadeler yerleştirilir ve dxdy yerine jakobiyan çarpı dudv yazılır."]},"beginTime":1997,"endTime":2086,"href":"/video/preview/7557418428767430028?parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL&text=Datch+Integral&t=1997&ask_summarization=1"}],"linkTemplate":"/video/preview/7557418428767430028?parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL&text=Datch+Integral&t=%%timestamp%%&ask_summarization=1"},"isAdultDoc":false,"relatedParams":{"text":"Calculus-II : Çift Katlı İntegralde Değişken Değiştirme (Change of 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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/2968351/d78706d1d3b142eecdeb98d896edc06c/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/shXExwAAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"18","reqid":"1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1769442543160788-653554166104779684-balancer-l7leveler-kubr-yp-sas-255-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. 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