Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.1.50b

Chapter 6, Problem 6.1.50b

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.

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Related Practice

Textbook Question

6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.

Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.

b. What is the height of a cylindrical shell at a point x in [0, 4]?

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Textbook Question

Displacement and distance from velocity Consider the velocity function shown below of an object moving along a line. Assume time is measured in seconds and distance is measured in meters. The areas of four regions bounded by the velocity curve and the t-axis are also given.

b. What is the displacement of the object over the interval [2, 6]?

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Textbook Question

55–58. Marginal cost Consider the following marginal cost functions.

b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.

C′(x)=200−0.05x

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Textbook Question

Distance traveled and displacement Suppose an object moves along a line with velocity (in ft/s) v(t)=6−2t, for 0≤t≤6, where t is measured in seconds.

b. Find the displacement of the object on the interval 0≤t≤6.

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Textbook Question

Work done by a spring A spring on a horizontal surface can be stretched and held 0.5 m from its equilibrium position with a force of 50 N.

b. How much work is done in compressing the spring 0.5 m from its equilibrium position?

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Textbook Question

55–58. Marginal cost Consider the following marginal cost functions.

b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.

C′(x) = 300+10x−0.01x²

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