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Her örnek için adım adım çözüm süreci ve grafikler üzerinden açıklamalar yapılmaktadır."]},"endTime":1995,"title":"İntegralin Uygulamaları Eğitim Videosu","beginTime":0}],"fullResult":[{"index":0,"title":"İntegralin Alan Uygulamaları","list":{"type":"unordered","items":["İntegralin ilk uygulaması, pozitif tanımlı f(x) fonksiyonunun x ekseni arasında kalan bölgenin alanını hesaplamaktır.","Pozitif tanımlı f(x) için alan, a'dan b'ye kadar f(x) dx integraline eşittir.","Negatif tanımlı f(x) için alan, a'dan b'ye kadar -f(x) dx integraline eşittir çünkü alan negatif olamaz."]},"beginTime":1,"endTime":93,"href":"/video/preview/3141441284461915231?parent-reqid=1769526030281427-8641223117059354046-balancer-l7leveler-kubr-yp-klg-269-BAL&text=Computer+Application+Integral+University&t=1&ask_summarization=1"},{"index":1,"title":"Örnek Sorular","list":{"type":"unordered","items":["Birinci örnek soruda, f(x) fonksiyonunun grafiği verilmiş ve -8'den 4'e kadar f(x) dx ve |f(x)| dx integrallerinin değerleri hesaplanmıştır.","İkinci örnek soruda, f(x) fonksiyonunun grafiği ve bazı alanlar verilmiş, -3'ten 4'e kadar f(x) dx integralinin değeri bulunmuştur.","Üçüncü örnek soruda, h(x) fonksiyonunun grafiği ve taralı bölgelerin alanları verilmiş, -5'ten 8'e kadar f(2x) dx integralinin değeri hesaplanmıştır."]},"beginTime":93,"endTime":334,"href":"/video/preview/3141441284461915231?parent-reqid=1769526030281427-8641223117059354046-balancer-l7leveler-kubr-yp-klg-269-BAL&text=Computer+Application+Integral+University&t=93&ask_summarization=1"},{"index":2,"title":"Değişken Değiştirme Yöntemi","list":{"type":"unordered","items":["Dördüncü örnek soruda, f(x) fonksiyonunun grafiği ve taralı alan verilmiş, -1'den 2'ye kadar f(x-2x²) dx integralinin değeri bulunmuştur.","Beşinci örnek soruda, f(x) fonksiyonu ve türevinin grafiği verilmiş, taralı bölgenin alanı hesaplanmıştır.","Altıncı örnek soruda, f(x) ve g(x) fonksiyonlarının grafikleri verilmiş, -4'ten 5'e kadar f(x)-g(x) dx integralinin değeri bulunmuştur."]},"beginTime":334,"endTime":502,"href":"/video/preview/3141441284461915231?parent-reqid=1769526030281427-8641223117059354046-balancer-l7leveler-kubr-yp-klg-269-BAL&text=Computer+Application+Integral+University&t=334&ask_summarization=1"},{"index":3,"title":"İki Eğri Arasındaki Alan","list":{"type":"unordered","items":["İki eğri arasındaki alanı bulmak için üst sınırdan alt sınırı çıkarmak yeterlidir.","İki eğri arasındaki alan, üstteki fonksiyonun alttaki fonksiyondan çıkarılmasıyla hesaplanır.","Örnek soruda, f(x) ve y=2x+4 doğrusu verilmiş, -4'ten 2'ye kadar f(x) dx ve 2'den 5'e kadar f(x) dx integrallerinin değerleri kullanılarak taralı bölgelerin toplamı hesaplanmıştır."]},"beginTime":502,"endTime":769,"href":"/video/preview/3141441284461915231?parent-reqid=1769526030281427-8641223117059354046-balancer-l7leveler-kubr-yp-klg-269-BAL&text=Computer+Application+Integral+University&t=502&ask_summarization=1"},{"index":4,"title":"İntegral Özellikleri ve Uygulamaları","list":{"type":"unordered","items":["Bir fonksiyonun birebir ve örten olması durumunda, belirli integral özellikleri kullanılarak taralı alanlar hesaplanabilir.","İntegralde alt ve üst sınırlara sabit değerler eklendiğinde, integral ifadesi değişerek taralı alanlar bulunabilir.","Çift fonksiyonların belirli integrali hesaplanırken değişken değiştirme yöntemi kullanılabilir."]},"beginTime":773,"endTime":999,"href":"/video/preview/3141441284461915231?parent-reqid=1769526030281427-8641223117059354046-balancer-l7leveler-kubr-yp-klg-269-BAL&text=Computer+Application+Integral+University&t=773&ask_summarization=1"},{"index":5,"title":"Rayman Toplamları","list":{"type":"unordered","items":["İntegralin kökeni Alman matematikçi Rayman'dır ve geometri ile bulamadığı eğrinin altındaki alanı hesaplamak için dikdörtgenler kullanmıştır.","Rayman alt toplamı, eğrinin altındaki alanı hesaplamak için dikdörtgenlerin sol köşelerinin eğri üzerinde olduğu toplam alanıdır.","Rayman üst toplamı, eğrinin altındaki alanı hesaplamak için dikdörtgenlerin sağ köşelerinin eğri üzerinde olduğu toplam alanıdır."]},"beginTime":999,"endTime":1201,"href":"/video/preview/3141441284461915231?parent-reqid=1769526030281427-8641223117059354046-balancer-l7leveler-kubr-yp-klg-269-BAL&text=Computer+Application+Integral+University&t=999&ask_summarization=1"},{"index":6,"title":"Rayman Toplamlarının Uygulamaları","list":{"type":"unordered","items":["Rayman alt toplamı ve üst toplamı hesaplanırken, aralık eşit parçalara bölünür ve her bir dikdörtgenin alanları toplanır.","Artan ve sürekli bir fonksiyonun belirli integrali, Rayman alt toplamı ve üst toplamı arasında bir değer alır.","Rayman alt toplamı, eğrinin altındaki alanın gerçek değerinden küçüktür."]},"beginTime":1201,"endTime":1505,"href":"/video/preview/3141441284461915231?parent-reqid=1769526030281427-8641223117059354046-balancer-l7leveler-kubr-yp-klg-269-BAL&text=Computer+Application+Integral+University&t=1201&ask_summarization=1"},{"index":7,"title":"İntegral Uygulamaları","list":{"type":"unordered","items":["Hız-zaman grafiğinde, eğrinin üzerindeki sağ üst köşelerle çizilen dikdörtgenlerin alanları, eğrinin altındaki alandan büyüktür ve bu alan 60 km² olarak verilmiştir.","Dikdörtgenlerin alanları toplandığında, eğrinin altındaki alandan büyük olan en küçük değer 301'dir.","İntegral hesaplamalarında, fonksiyonun grafiğini doğru görmek önemlidir ve bazı durumlarda ortak çarpanlar alınarak fonksiyonun türevi 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almaktadır."]},"beginTime":1655,"endTime":1754,"href":"/video/preview/3141441284461915231?parent-reqid=1769526030281427-8641223117059354046-balancer-l7leveler-kubr-yp-klg-269-BAL&text=Computer+Application+Integral+University&t=1655&ask_summarization=1"},{"index":9,"title":"İntegral Uygulamaları","list":{"type":"unordered","items":["İki eğri arasındaki alan hesaplanırken, üst eğrinin altındaki eğriden çıkarılması gerekir.","Depremde fay uzunluğu, depremin şiddeti ile çıkan enerjinin çarpımına eşittir ve bu ilişki integral ile ifade edilebilir.","Düşey atış yapabilen bir tankın attığı top mermisinin hız fonksiyonu v(t) = 120 - 40t şeklindedir ve top mermisi dağın zirve noktasına kadar çıkıp 6 saniyede yere düştüğünde, dağın yüksekliği 180 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İlk bölümde parçalı fonksiyonların integrali, mutlak değer fonksiyonlarının integrali ve belirli integralin türevi konuları açıklanırken, ikinci bölümde bu konular tekrarlanmaktadır. 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