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

التنفس عملية دورية، حيث تستمر كل دورة تنفسية كاملة حوالي 5 ثوان.الهدف هو تحديد حجم الهواء المستنشق في الرئتين في أي وقت خلال هذه الدورة.يتغير معدل تدفق الهواء مع مرور الوقت ويمكن نمذجته باستخدام دالة جيبية. يظهر أن تدفق الهواء يبلغ ذروته أثناء منتصف الشهيق وينعكس أثناء الزفير، بمعدل أقصى يبلغ حوالي 0.5 لتر/ثانية.يتراكم الحجم اللانهائي الصغير dV خلال وقت قصير dt إلى الحجم الإجمالي المستنشق عن طريق تكامل معدل تدفق الهواء من 0 إلى t.طريقة الاستبدال تبسط التكامل عن طريق تعيين المتغير u على 2πx على 5. يعبر عن التفاضل dx بحسب du ليطابق المتغير الجديد. تتغير حدود التكامل أيضا لتتوافق مع المتغير الجديد، حيث تتراوح من 0 إلى 2πt فوق 5.يصبح التكامل صيغة جيبية قياسية، تكاملها جيب تمام سالب، ويقيم باستخدام حدود التكامل.هذا يعطي حجم الهواء المستنشق باللترات كدالة للوقت خلال دورة التنفس، ويصل إلى أقصى حد أثناء الاستنشاق.يشمل هذا النموذج كلا من اتجاه ومعدل تدفق الهواء، مما يعطي رؤية واضحة لحجم التنفس مع مرور الوقت.

تطبيق التكامل: حل المشكلات

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&#x625;&#x644;&#x649; &#x627;&#x644;&#x631;&#x626;&#x62A;&#x64A;&#x646;.&#x627;&#x644;&#x646;&#x645;&#x648;&#x630;&#x62C; &#x627;&#x644;&#x62C;&#x64A;&#x628;&#x64A; &#x644;&#x62A;&#x62F;&#x641;&#x642; &#x627;&#x644;&#x647;&#x648;&#x627;&#x621;&#x644;&#x627; &#x64A;&#x643;&#x648;&#x646; &#x62A;&#x62F;&#x641;&#x642; &#x627;&#x644;&#x647;&#x648;&#x627;&#x621; &#x623;&#x62B;&#x646;&#x627;&#x621; &#x627;&#x644;&#x62A;&#x646;&#x641;&#x633; &#x62B;&#x627;&#x628;&#x62A;&#x64B;&#x627;&#x61B; &#x628;&#x644; &#x64A;&#x62A;&#x63A;&#x64A;&#x631; &#x645;&#x639; &#x627;&#x644;&#x632;&#x645;&#x646;&#x60C; &#x648;&#x64A;&#x645;&#x643;&#x646; &#x62A;&#x645;&#x62B;&#x64A;&#x644;&#x647; &#x628;&#x62F;&#x642;&#x629; &#x623;&#x643;&#x628;&#x631; &#x628;&#x62F;&#x627;&#x644;&#x629; &#x62C;&#x64A;&#x628;&#x64A;&#x629;. &#x641;&#x64A; &#x62F;&#x648;&#x631;&#x629; &#x62A;&#x646;&#x641;&#x633; &#x637;&#x628;&#x64A;&#x639;&#x64A;&#x629;&#x60C; &#x64A;&#x628;&#x644;&#x63A; &#x645;&#x639;&#x62F;&#x644; &#x62A;&#x62F;&#x641;&#x642; &#x627;&#x644;&#x647;&#x648;&#x627;&#x621; &#x642;&#x64A;&#x645;&#x629; &#x639;&#x638;&#x645;&#x649; &#x62A;&#x642;&#x627;&#x631;&#x628; 0.5 L/s &#x639;&#x646;&#x62F; &#x645;&#x646;&#x62A;&#x635;&#x641; &#x627;&#x644;&#x634;&#x647;&#x64A;&#x642;&#x60C; &#x62B;&#x645; &#x64A;&#x62A;&#x646;&#x627;&#x642;&#x635; &#x628;&#x634;&#x643;&#x644; &#x645;&#x62A;&#x646;&#x627;&#x638;&#x631; &#x62D;&#x62A;&#x649; &#x64A;&#x635;&#x644; &#x625;&#x644;&#x649; &#x627;&#x644;&#x635;&#x641;&#x631; &#x639;&#x646;&#x62F; &#x628;&#x62F;&#x627;&#x64A;&#x629; &#x643;&#x644; &#x645;&#x631;&#x62D;&#x644;&#x629; &#x648;&#x646;&#x647;&#x627;&#x64A;&#x62A;&#x647;&#x627;. &#x64A;&#x645;&#x643;&#x646; &#x646;&#x645;&#x630;&#x62C;&#x629; &#x647;&#x630;&#x627; &#x627;&#x644;&#x633;&#x644;&#x648;&#x643; &#x627;&#x644;&#x62F;&#x648;&#x631;&#x64A; &#x628;&#x627;&#x644;&#x62F;&#x627;&#x644;&#x629; &#x627;&#x644;&#x62A;&#x627;&#x644;&#x64A;&#x629;:&#xA0;\begin{equation*}\jfrac{dV}{dt} = 0.5 \sin\liparens{\jfrac{2\pi t}{5}}\end{equation*}&#x62D;&#x64A;&#x62B; dV/dt &#x647;&#x648; &#x645;&#x639;&#x62F;&#x644; &#x62A;&#x62F;&#x641;&#x642; &#x627;&#x644;&#x647;&#x648;&#x627;&#x621; &#x628;&#x627;&#x644;&#x644;&#x62A;&#x631; &#x641;&#x64A; &#x627;&#x644;&#x62B;&#x627;&#x646;&#x64A;&#x629;&#x60C; &#x648; t &#x647;&#x648; &#x627;&#x644;&#x632;&#x645;&#x646; &#x628;&#x627;&#x644;&#x62B;&#x648;&#x627;&#x646;&#x64A;.&#x627;&#x644;&#x62A;&#x643;&#x627;&#x645;&#x644; &#x628;&#x627;&#x633;&#x62A;&#x62E;&#x62F;&#x627;&#x645; &#x627;&#x644;&#x62A;&#x639;&#x648;&#x64A;&#x636;&#x644;&#x62A;&#x62D;&#x62F;&#x64A;&#x62F; &#x62D;&#x62C;&#x645; &#x627;&#x644;&#x647;&#x648;&#x627;&#x621; &#x627;&#x644;&#x645;&#x633;&#x62A;&#x646;&#x634;&#x64E;&#x642; V(t) &#x639;&#x646;&#x62F; &#x623;&#x64A; &#x632;&#x645;&#x646; t&#x60C; &#x64A;&#x62A;&#x645; &#x62A;&#x643;&#x627;&#x645;&#x644; &#x62F;&#x627;&#x644;&#x629; &#x62A;&#x62F;&#x641;&#x642; &#x627;&#x644;&#x647;&#x648;&#x627;&#x621; &#x639;&#x644;&#x649; &#x627;&#x644;&#x646;&#x62D;&#x648; &#x627;&#x644;&#x622;&#x62A;&#x64A;:&#xA0;\begin{equation*}V(t) = \int_0^t 0.5 \sin\liparens{\jfrac{2\pi x}{5}}dx\end{equation*}&#x648;&#x628;&#x627;&#x633;&#x62A;&#x62E;&#x62F;&#x627;&#x645; &#x627;&#x644;&#x62A;&#x639;&#x648;&#x64A;&#x636; u = 2&#x3C0;x/5&#x60C; &#x64A;&#x62A;&#x628;&#x633;&#x637; &#x627;&#x644;&#x62A;&#x643;&#x627;&#x645;&#x644; &#x625;&#x644;&#x649;:\begin{equation*}V(t) = \jfrac{5}{2\pi} \int_0^{\jfrac{2\pi t}{5}} 0.5 \sin(u) du\end{equation*}&#x648;&#x64A;&#x643;&#x648;&#x646; &#x646;&#x627;&#x62A;&#x62C; &#x647;&#x630;&#x627; &#x627;&#x644;&#x62A;&#x643;&#x627;&#x645;&#x644;:&#xA0;\begin{equation*}V(t) = -\jfrac{5}{4\pi} \left[\cos\left(\jfrac{2\pi t}{5}\right) - 1\right]\end{equation*}&#x62A;&#x635;&#x641; &#x647;&#x630;&#x647; &#x627;&#x644;&#x62F;&#x627;&#x644;&#x629; &#x62D;&#x62C;&#x645; &#x627;&#x644;&#x647;&#x648;&#x627;&#x621; &#x627;&#x644;&#x645;&#x633;&#x62A;&#x646;&#x634;&#x642; &#x628;&#x645;&#x631;&#x648;&#x631; &#x627;&#x644;&#x632;&#x645;&#x646;&#x60C; &#x648;&#x62A;&#x62D;&#x62F;&#x62F; &#x645;&#x642;&#x62F;&#x627;&#x631; &#x62A;&#x62F;&#x641;&#x642; &#x627;&#x644;&#x647;&#x648;&#x627;&#x621; &#x648;&#x627;&#x62A;&#x62C;&#x627;&#x647;&#x647;. &#x64A;&#x628;&#x644;&#x63A; 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&#x62A;&#x62A;&#x636;&#x645;&#x646; &#x639;&#x645;&#x644;&#x64A;&#x629; &#x627;&#x644;&#x62A;&#x646;&#x641;&#x633; &#x625;&#x62F;&#x62E;&#x627;&#x644; ...

application-of-integration-problem-solving

Education

JoVE Core

Calculus

حساب التفاضل والتكامل

Unit

وحدة

Integrals

التكاملات

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

Application of Integration: Problem Solving

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

area problem

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properties of definite integral iii

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application of integration: problem solving

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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