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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.1.13b
Chapter 6, Problem 6.1.13b

13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.


b. Find the displacement over the given interval. 


v(t) = 3t²−6t on [0, 3]

Verified step by step guidance
1
Identify the given velocity function: \(v(t) = 3t^{2} - 6t\) and the time interval \([0, 3]\) seconds.
Recall that displacement over a time interval \([a, b]\) is found by integrating the velocity function over that interval: \(\text{Displacement} = \int_{a}^{b} v(t) \, dt\).
Set up the definite integral for displacement: \(\int_{0}^{3} (3t^{2} - 6t) \, dt\).
Integrate the function term-by-term: the integral of \$3t^{2}\( is \)t^{3}\(, and the integral of \)-6t\( is \)-3t^{2}\(, so the antiderivative is \)t^{3} - 3t^{2}$.
Evaluate the antiderivative at the bounds and subtract: calculate \(\left[ t^{3} - 3t^{2} \right]_{0}^{3} = (3^{3} - 3 \times 3^{2}) - (0 - 0)\) to find the displacement.

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Key Concepts

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

Velocity and Displacement Relationship

Velocity represents the rate of change of position with respect to time. Displacement over a time interval is the net change in position, which can be found by integrating the velocity function over that interval. This integral sums the instantaneous velocities to give the overall change in position.
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Definite Integral as Net Area

The definite integral of a velocity function over a time interval calculates the net area between the velocity curve and the time axis. Positive areas correspond to movement in one direction, while negative areas represent movement in the opposite direction. The sum of these areas gives the total displacement.
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Evaluating Definite Integrals

To find displacement, compute the definite integral of v(t) from the start to end time. This involves finding the antiderivative of v(t), then applying the Fundamental Theorem of Calculus by subtracting the antiderivative evaluated at the lower limit from that at the upper limit.
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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]?

Textbook Question

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.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


b. If a region is revolved about the y-axis, then the shell method must be used.

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Textbook Question

40–43. Population growth


When records were first kept (t=0), the population of a rural town was 250 people. During the following years, the population grew at a rate of P′(t) = 30(1+√t), where t is measured in years.


b. Find the population P(t) at any time t≥0.

Textbook Question

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.

Textbook Question

Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.


b. Suppose P=10, A=20, and r=0. If the initial population is N(0)=100, does the population ever become extinct? Explain.