Name: integration in breathing cycle modeling
Uploaded: 2026-01-12T12:24:28.000Z
Duration: 1 min 30 s
Channel: JoVE

Nefes almak, döngüsel bir süreçtir ve her tam solunum döngüsü yaklaşık 5 saniye sürer.Amaç, bu döngü boyunca akciğerlerde herhangi bir zamanda solunan hava hacmini bulmaktır.Hava akışının hızı zamanla değişir ve sinüzoidal bir fonksiyon kullanılarak modellenebilir. Hava akışının orta nefes alma sırasında zirve olduğu, nefes verirken ise tersine döndüğünü, maksimum hızın yaklaşık 0,5 L/s olduğunu gösterir.Kısa bir sürede dV, dt'nin sonsuz küçük hacmi dt, 0'dan t'ye hava akışı hızını entegre ederek toplam solunma hacmine biriktirir.Yerine geçme yöntemi, u değişkenini 5üzerinde 2 πx olarak ayarlayarak integralı basitleştirir. Diferansiyel dx , yeni değişkenle eşleşmesi için du cinsinden ifade edilir. Entegrasyon sınırları da yeni değişkene uyacak şekilde değişir ve 0'dan 2πt'ye 5'e kadar değişir.İnintegral, integral sınırları kullanılarak değerlendirilen negatif kosinüs olan standart bir sinüs formu haline gelir.Bu, soluma döngüsündeki zamanın bir fonksiyonu olarak litrede solunan hava hacmini verir ve soluma sırasında maksimuma ulaşır.Bu model, hava akışının hem yönünü hem de hızını içerir ve zaman içinde solunum hacminin net bir görüntüsünü sağlar.

İntegralin uygulanması: problem Çözümü

Application of Integration: Problem Solving

Solunum s&#xFC;reci, solunum d&#xF6;ng&#xFC;s&#xFC; olarak adland&#x131;r&#x131;lan ve genellikle yakla&#x15F;&#x131;k be&#x15F; saniye s&#xFC;ren, havan&#x131;n periyodik olarak al&#x131;nmas&#x131; ve d&#x131;&#x15F;ar&#x131; at&#x131;lmas&#x131;n&#x131; kapsar. Akci&#x11F;erlere al&#x131;nan hava hacminin zamana ba&#x11F;l&#x131; bir fonksiyon olarak modellenmesi, pulmoner ventilasyonun dinamikleri ve verimlilik hakk&#x131;nda &#xF6;nemli bilgiler sunar. Bu hacim, akci&#x11F;erlere giren havan&#x131;n k&#xFC;m&#xFC;latif etkisini yans&#x131;tan hava ak&#x131;&#x15F; h&#x131;z&#x131;n&#x131;n zamana g&#xF6;re integralinin al&#x131;nmas&#x131;yla belirlenir.Sin&#xFC;zoidal Hava Ak&#x131;&#x15F;&#x131; ModeliSolunum s&#x131;ras&#x131;nda hava ak&#x131;&#x15F;&#x131; sabit de&#x11F;ildir; zaman i&#xE7;inde de&#x11F;i&#x15F;kenlik g&#xF6;sterir ve en do&#x11F;ru bi&#xE7;imde sin&#xFC;zoidal bir fonksiyonla temsil edilir. Normal bir solunum d&#xF6;ng&#xFC;s&#xFC;nde hava ak&#x131;&#x15F; h&#x131;z&#x131;, inhalasyonun (inspirasyon) orta noktas&#x131;nda yakla&#x15F;&#x131;k 0,5 L/s d&#xFC;zeyinde maksimum de&#x11F;ere ula&#x15F;&#x131;r ve her bir faz&#x131;n ba&#x15F;lang&#x131;c&#x131;nda ve sonunda simetrik olarak s&#x131;f&#x131;ra do&#x11F;ru azal&#x131;r. Bu periyodik davran&#x131;&#x15F; a&#x15F;a&#x11F;&#x131;daki fonksiyonla modellenebilir:\begin{equation*}\jfrac{dV}{dt} = 0.5 \sin\liparens{\jfrac{2\pi t}{5}}\end{equation*}Burada dV/dt, saniyede litre cinsinden hava ak&#x131;&#x15F; h&#x131;z&#x131;n&#x131;, t ise saniye cinsinden zaman&#x131; ifade eder.Yerine Koyma (&#x130;kame) Y&#xF6;ntemi ile &#x130;ntegrasyonHerhangi bir t an&#x131;nda inhalasyonla al&#x131;nan hava hacmi V(t)&#x2019;yi belirlemek i&#xE7;in hava ak&#x131;&#x15F; fonksiyonu a&#x15F;a&#x11F;&#x131;daki &#x15F;ekilde integrallenir:\begin{equation*}V(t) = \int_0^t 0.5 \sin\liparens{\jfrac{2\pi x}{5}}dx\end{equation*}u = 2&#x3C0;x/5 ikamesi yap&#x131;ld&#x131;&#x11F;&#x131;nda integral a&#x15F;a&#x11F;&#x131;daki bi&#xE7;imde sadele&#x15F;tirilir:\begin{equation*}V(t) = \jfrac{5}{2\pi} \int_0^{\jfrac{2\pi t}{5}} 0.5 \sin(u) du\end{equation*}Bu integral a&#x15F;a&#x11F;&#x131;daki de&#x11F;eri al&#x131;r:\begin{equation*}V(t) = -\jfrac{5}{4\pi} \left[\cos\left(\jfrac{2\pi t}{5}\right) - 1\right]\end{equation*}Bu fonksiyon, hava ak&#x131;&#x15F;&#x131;n&#x131;n hem b&#xFC;y&#xFC;kl&#xFC;&#x11F;&#xFC;n&#xFC; hem de i&#x15F;aretini dikkate alarak zaman i&#xE7;inde inhalasyonla al&#x131;nan hava hacmini tan&#x131;mlar. Model, tipik bir solunum d&#xF6;ng&#xFC;s&#xFC; s&#x131;ras&#x131;nda akci&#x11F;erlerin fizyolojik davran&#x131;&#x15F;&#x131;n&#x131; yans&#x131;tarak, inhalasyonun orta a&#x15F;amas&#x131;nda maksimum de&#x11F;ere ula&#x15F;&#x131;r.

İntegralin Uygulanması: Problem Çözümü

Solunum s&#xFC;reci, solunum d&#xF6;ng&#xFC;s&#xFC; olarak adland&#x131;r&#x131;lan ve genellikle yakla&#x15F;&#x131;k be&#x15F; saniye s&#xFC;ren, ...

application-of-integration-problem-solving

Education

JoVE Core

Calculus

Unit

Integrals

KEY TERMS AND CONCEPTS

integration in breathing cycle modeling

Integration can be used to model the volume of air inhaled during breathing. The respiratory cycle is the repeating intake and expulsion of air, and it usually lasts about five seconds. By treating airflow as a function of time, students can see how the amount of air in the lungs changes during a breath.

Airflow during respiration is not constant. It changes in a smooth, wave-like pattern, so a sinusoidal function is a good model. In a normal breathing cycle, the airflow rate reaches a maximum of about 0.5 liters per second at the midpoint of inhalation. It then falls symmetrically to zero at the start and end of each phase.

To find the inhaled volume V(t), the airflow rate is integrated over time. The model uses dV/dt in liters per second and time t in seconds. With the substitution u = 2πx/5, the integral becomes easier to solve. The result gives a function for the volume of inhaled air over time.

This integral describes how much air enters the lungs as breathing continues. It also captures both the size and direction of airflow. The volume curve peaks at mid-inhalation, which matches the normal pattern of lung movement during a typical breath cycle.

Integration in Breathing Cycle Modeling

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Integration can be used to model the volume of air inhaled during breathing. The respiratory cycle is the repeating intake and expulsion of air, and it usually lasts about five seconds. By treating airflow as a function of time, students can see how the amount of air in the lungs changes during a breath. Airflow during respiration is not constant. It changes in a smooth, wave-like pattern, so a sinusoidal function is a good model. In a normal breathing cycle, the airflow rate reaches a maximum...

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key terms and concepts

See how integration models inhaled air in the breathing cycle using a sinusoidal airflow function, substitution, and a volume curve for lung ventilation.

Integration in Breathing Cycle Modeling | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Integration in Breathing Cycle Modeling

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