Blood flow A typical human heart pumps 70 mL of blood (the stroke volume) with each beat. Assuming a heart rate of 60 beats/min (1 beat/s), a reasonable model for the outflow rate of the heart is V′(t)=70(1+sin 2πt), where V(t) is the amount of blood (in milliliters) pumped over the interval [0,t],V(0)=0 and t is measured in seconds.

b. Find the function that gives the total blood pumped between t=0 and a future time t>0.

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Identify the given rate of blood flow as the derivative of the volume function: \(V'(t) = 70(1 + \sin(2\pi t))\) milliliters per second.

Recall that to find the total volume pumped over time, you need to find the original function \(V(t)\) by integrating the rate function \(V'(t)\) with respect to \(t\).

Set up the integral: \(V(t) = \int 70(1 + \sin(2\pi t)) \, dt\).

Split the integral into two parts for easier integration: \(V(t) = 70 \int 1 \, dt + 70 \int \sin(2\pi t) \, dt\).

Integrate each part separately, remembering to include the constant of integration \(C\), and then use the initial condition \(V(0) = 0\) to solve for \(C\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Understanding the Derivative as a Rate of Change

The derivative V′(t) represents the instantaneous rate at which blood is pumped by the heart at time t. Recognizing that V′(t) is the rate of change of the total volume V(t) is essential to relate the given function to the total blood pumped over time.

Definite Integration to Find Accumulated Quantity

To find the total blood pumped from time 0 to t, we integrate the rate function V′(t) over [0, t]. Integration accumulates the instantaneous rates over time, yielding the total volume V(t). This process reverses differentiation and uses the initial condition to find the constant of integration.

Applying Initial Conditions to Determine the Integration Constant

Since V(0) = 0, the initial amount of blood pumped is zero. After integrating V′(t), we use this initial condition to solve for the constant of integration, ensuring the function V(t) accurately models the total blood volume pumped starting from zero.