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Not bad. 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Eğitmen, integral konusunun farklı uygulamalarını açıklamaktadır.","Videoda integralin sadece alan ve hacim hesaplamalarında değil, eğrinin uzunluğunu da bulmada nasıl kullanılabileceği anlatılmaktadır. Eğitmen önce formülün matematiksel temellerini göstererek, eğriyi eşit parçalara ayırma yöntemini, dik üçgen kullanarak uzunluk formülünü ve türevin tanımını kullanarak formülleri türetmeyi adım adım anlatır, ardından integral notasyonuna dönüştürme sürecini açıklar.","Video, f(x) = 1/3x³ + 1/4x fonksiyonunun 1 ile 3 aralığındaki uzunluğunu hesaplama örneği üzerinden konuyu pekiştirir ve bir sonraki derste başka bir konseptin anlatılacağı bilgisiyle sonlanır."]},"endTime":1412,"title":"İntegral ile Eğrinin Uzunluğunu Hesaplama","beginTime":0}],"fullResult":[{"index":0,"title":"İntegralin Eğrinin Uzunluğunu Bulma Kullanımı","list":{"type":"unordered","items":["İntegral daha önce eğrinin altında kalan alan ve hacim hesaplamalarında kullanılmıştır.","Bu videoda integralin bir eğrinin uzunluğunu bulma yöntemi incelenecektir.","Eğrinin uzunluğu, eğrinin üzerindeki noktaların arasındaki mesafelerin toplamıdır."]},"beginTime":0,"endTime":57,"href":"/video/preview/9011499757430886225?parent-reqid=1769429006763855-12099915074130680452-balancer-l7leveler-kubr-yp-sas-204-BAL&text=integral+evolution&t=0&ask_summarization=1"},{"index":1,"title":"Eğrinin Uzunluğunu Hesaplama Yöntemi","list":{"type":"unordered","items":["Eğrinin uzunluğu hesaplanırken, aralık eşit parçalara (delta x) bölünür ve her parçada oluşan doğru parçalarının uzunlukları toplanır.","Parçaların sayısı arttıkça yaklaşım daha doğru olur ve delta x sıfıra giderken kesin uzunluk bulunur.","Her bir doğru parçasının uzunluğu, dik üçgen yardımıyla hesaplanabilir."]},"beginTime":57,"endTime":238,"href":"/video/preview/9011499757430886225?parent-reqid=1769429006763855-12099915074130680452-balancer-l7leveler-kubr-yp-sas-204-BAL&text=integral+evolution&t=57&ask_summarization=1"},{"index":2,"title":"Doğru Parçasının Uzunluğunu Hesaplama","list":{"type":"unordered","items":["Doğru parçasının uzunluğu (lk), dik üçgenin hipotenüsü olarak hesaplanır.","Dik üçgenin taban uzunluğu delta x, dik kenarlar ise f(x_k) ve f(x_k-1) değerleridir.","Dik üçgenin hipotenüsü, kök içinde delta x² + (f(x_k) - f(x_k-1))² formülüyle hesaplanır."]},"beginTime":238,"endTime":402,"href":"/video/preview/9011499757430886225?parent-reqid=1769429006763855-12099915074130680452-balancer-l7leveler-kubr-yp-sas-204-BAL&text=integral+evolution&t=238&ask_summarization=1"},{"index":3,"title":"Türevin Tanımı ve Eğrinin Uzunluğu","list":{"type":"unordered","items":["f(x_k) - f(x_k-1) ifadesi, delta x'e bölündüğünde fonksiyonun x_k noktasındaki türevini verir.","Türevin tanımı, limit delta x sıfıra giderken çıktılar arası fark bölü girdiler arası fark şeklinde ifade edilir.","Doğru parçasının uzunluğu, türev tanımı kullanılarak lk = kök içinde (delta x² + (f'(x_k) × delta x)²) formülüyle hesaplanabilir."]},"beginTime":402,"endTime":696,"href":"/video/preview/9011499757430886225?parent-reqid=1769429006763855-12099915074130680452-balancer-l7leveler-kubr-yp-sas-204-BAL&text=integral+evolution&t=402&ask_summarization=1"},{"index":4,"title":"Eğrinin Uzunluğu Formülü","list":{"type":"unordered","items":["Eğrinin uzunluğu hesaplanırken, her bir doğru parçasının uzunluğu (lk) hesaplanıp toplanır.","Her bir doğru parçasının uzunluğu, delta x çarpı kök içinde (1 + f'(x)k)² formülüyle hesaplanır.","Parçaların sayısı (n) sonsuza giderken, bu yaklaşım eğrinin gerçek uzunluğunu temsil eder ve hata payı sıfıra iner."]},"beginTime":709,"endTime":872,"href":"/video/preview/9011499757430886225?parent-reqid=1769429006763855-12099915074130680452-balancer-l7leveler-kubr-yp-sas-204-BAL&text=integral+evolution&t=709&ask_summarization=1"},{"index":5,"title":"İntegral Formülünün Oluşturulması","list":{"type":"unordered","items":["Eğrinin uzunluğu formülü integral notasyonuna oturtularak, ∫√(1 + (f'(x))²) dx şeklinde ifade edilir.","Bu formül, f(x) eğrisinin [a,b] aralığındaki birebir uzunluğunu verir.","Y eksenine paralel eğriler için de benzer formül kullanılır: ∫√(1 + (g'(y))²) dy."]},"beginTime":872,"endTime":1008,"href":"/video/preview/9011499757430886225?parent-reqid=1769429006763855-12099915074130680452-balancer-l7leveler-kubr-yp-sas-204-BAL&text=integral+evolution&t=872&ask_summarization=1"},{"index":6,"title":"Örnek Soru Çözümü","list":{"type":"unordered","items":["f(x) = 1/3x³ + 1/4x eğrisinin [1,3] aralığındaki uzunluğu hesaplanacak.","Önce f'(x) türevi bulunur: f'(x) = 1/3x² - 1/4x⁻².","f'(x)² ifadesi hesaplanır ve kök dışına çıkarılır: √(x⁴ + 1/2 + 1/16x⁴)."]},"beginTime":1008,"endTime":1290,"href":"/video/preview/9011499757430886225?parent-reqid=1769429006763855-12099915074130680452-balancer-l7leveler-kubr-yp-sas-204-BAL&text=integral+evolution&t=1008&ask_summarization=1"},{"index":7,"title":"İntegralin 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Eğitmen, evrişim formülünü açıklayarak başlayıp, sinyallerin dikey eksen etrafında katlanması ve zaman kaydırılması gibi kavramları açıklamaktadır. Ardından, evrişim integralinin farklı zaman aralıklarında nasıl hesaplanacağı detaylı olarak gösterilmekte ve son olarak, hesaplanan evrişim fonksiyonunun grafiği çizilmektedir. Video, doğrusal zamanla değişmez sistemler için evrişim işleminin nasıl uygulanacağını göstermektedir."]},"endTime":1159,"title":"Sinyal Evrişimi Hesaplama Dersi","beginTime":0}],"fullResult":[{"index":0,"title":"Sinyallerin Evrişimi","list":{"type":"unordered","items":["İki sinyalin evrişimi yt = x(t) * h(t) şeklinde ifade edilir ve integral olarak -∞'den +∞'ye x(t) * h(t) dt veya değişme özelliği kullanılarak h(t) * x(t) dt şeklinde hesaplanabilir.","h(-t) fonksiyonu dikey eksen etrafında katlanması anlamına gelirken, h(t-τ) fonksiyonu fonksiyonda τ değeri yerine t-τ yazarak sağa t süre kaydırma anlamına gelir.","Evrişim işlemi doğrusal zamanla değişmez sistemler için geçerlidir ve zamanla değişmezlik özelliği kullanılarak, evrişim işlemi yapıldıktan sonra sonucu iki birim sola kaydırarak doğru sonuç elde 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