4.13
不定積分
貯水タンクへの水の流入流量は一定ではなく、時間の経過とともに増加します。当初、ポンプは毎分5リットルの流入流量で水を供給します。しかし、圧力上昇やシステム調整により、流入流量は1分あたり2リットルずつ増加します。このシナリオは、数学的には線形関数で記述できます。
\begin{equation}
f(t…
水は大型の工業用貯蔵タンクに流れ込みますが、流入速度は一定ではありません。最初はポンプが5 m3/sで水を供給します。
システムの継続的な調整により、水の流入量は恒定して毎秒2立方メートルで増加しています。
この一定の変化により、水の流入速度は時間の線形関数となり、2は増加率を示す傾き、5は初期流入速度を示します。
目標は、任意の時点でタンク内に蓄えられた水の総量を見つけることです。これには無限積分が必要であり、時間経過による水の流入速度を積分して水の総体積を求める。
特定の瞬間の流入を分析する代わりに、積分は時間に関する水の総体積を単一の関数で表現します。
線形流入速度を積分すると二次項が導入され、これは水量の加速する増加を示し、線形項は初期流入量からの一定の成長を示します。
最後に、積分定数は t の体積関数がゼロに等しいことで与えられ、これはタンク内の初期水の体積に対応します。
View the full transcript and gain access to JoVE Core videos
Q1: What is an indefinite integral and why is it used?
An indefinite integral finds the total accumulation of a quantity by integrating its rate of change over time. Instead of analyzing rates at specific instants, indefinite integration provides a single function describing the total amount at any moment. For example, integrating a water inflow rate function yields the total volume stored in a tank as a function of time.
Q2: How does the power rule apply when integrating a linear inflow rate?
When integrating a linear inflow rate function like 5 + 2t, the power rule generates both quadratic and linear terms. The quadratic term t² represents accelerating growth from the increasing inflow rate, while the linear term 5t represents steady growth from the initial inflow. This combination shows how water volume accumulates over time.
Q3: What does the constant of integration represent in a water storage problem?
The constant of integration represents the initial volume of water present in the tank before the inflow process begins. It is determined by evaluating the volume function at t equals zero. This constant ensures the indefinite integral accurately describes the total water volume at any given time.
Q4: Why does integrating a changing inflow rate produce a quadratic term?
A quadratic term emerges because the inflow rate itself changes linearly over time. Integrating a linear rate of change produces a quadratic function, reflecting how water accumulation accelerates as the pump delivers more water per unit time. This quadratic pattern demonstrates the compounding effect of an increasing inflow rate.
Q5: How can indefinite integration solve real-world accumulation problems?
Indefinite integration converts a rate function into a total quantity function, enabling prediction of accumulation at any time. By integrating the inflow rate, engineers obtain a comprehensive volume equation V(t) that describes total water storage without calculating discrete time intervals. This approach applies to growth models with integration problem solving across engineering and science.
Q6: What is the relationship between inflow rate and total volume in the tank?
The inflow rate is the derivative of total volume with respect to time. Integrating the inflow rate function recovers the total volume function. In the water tank example, the linear inflow rate 5 + 2t integrates to produce V(t) = 5t + t² + C, where each term corresponds to a component of the inflow.
Q7: How does the initial pump rate affect the integrated volume function?
The initial pump rate of 5 m³/s becomes the coefficient of the linear term in the integrated volume function. This linear term 5t represents the constant contribution to total volume from the initial inflow rate throughout the time period. The increasing rate component then adds the quadratic term, showing how total volume grows faster over time.