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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 6 - Applications of IntegrationProblema 6.1.12b
Capitolo 6, Problema 6.1.12b

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.

Guida verificata passo dopo passo
1
Recall that displacement over a time interval is the net change in position, which can be found by integrating the velocity function over that interval.
Set up the definite integral for displacement using the given velocity function \(v(t) = 6 - 2t\) over the interval \(0 \leq t \leq 6\): \[\int_0^6 (6 - 2t) \, dt\]
Integrate the function \(6 - 2t\) with respect to \(t\). The integral of \(6\) is \$6t$, and the integral of $-2t$ is $-t^2$.
Evaluate the antiderivative at the upper and lower limits of the interval, \(t=6\) and \(t=0\), and subtract to find the displacement: \[\left[6t - t^2\right]_0^6 = (6 \times 6 - 6^2) - (6 \times 0 - 0^2)\]
Simplify the expression to find the net displacement of the object over the time interval.

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Velocity and Displacement

Velocity is the rate of change of position with respect to time and can be positive or negative, indicating direction. Displacement is the net change in position over a time interval and is found by integrating the velocity function over that interval.
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10:17
Using The Velocity Function

Definite Integral as Net Change

The definite integral of a velocity function from time a to b gives the net change in position, or displacement, during that time. It sums the signed areas under the velocity curve, accounting for direction.
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05:43
Definition of the Definite Integral

Evaluating Definite Integrals

To find displacement, compute the definite integral of v(t) from 0 to 6 by finding an antiderivative and applying the Fundamental Theorem of Calculus. This involves substituting the limits into the antiderivative and subtracting.
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05:43
Definition of the Definite Integral
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Domanda del libro di testo

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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Domanda del libro di testo

For the given regions R₁ and R₂, complete the following steps.


b. Find the area of region R₂ using geometry and the answer to part (a).


R₁is the region in the first quadrant bounded by the line x=1 and the curve y=6x(2−x^2)^2; R₂ is the region in the first quadrant bounded the curve y=6x(2−x^2)^2and the line y=6x.

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Domanda del libro di testo

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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Domanda del libro di testo

Compressing and stretching a spring Suppose a force of 15 N is required to stretch and hold a spring 0.25 m from its equilibrium position.

b. How much work is required to compress the spring 0.2 m from its equilibrium position?

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Domanda del libro di testo

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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Domanda del libro di testo

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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