Skip to main content
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 6.1.24b
Capitolo 6, Problema 6.1.24b

Cycling distance A cyclist rides down a long straight road with a velocity (in m/min) given by v(t) = 400−20t, for 0≤t≤10, where t is measured in minutes.


b. How far does the cyclist travel in the first 10 min?

Guida verificata passo dopo passo
1
Identify the velocity function given: \(v(t) = 400 - 20t\), where \(t\) is in minutes and \(v(t)\) is in meters per minute.
Recall that the distance traveled over a time interval can be found by integrating the velocity function over that interval. So, the distance \(D\) traveled from \(t=0\) to \(t=10\) is given by the definite integral \(D = \int_0^{10} v(t) \, dt\).
Set up the integral explicitly: \(D = \int_0^{10} (400 - 20t) \, dt\).
Integrate the function term-by-term: the integral of \(400\) with respect to \(t\) is \$400t$, and the integral of $-20t$ with respect to $t$ is $-10t^2$.
Evaluate the resulting expression \(400t - 10t^2\) at the upper limit \(t=10\) and subtract the value at the lower limit \(t=0\) to find the total distance traveled.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
2m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Velocity as a Function of Time

Velocity describes the rate of change of position with respect to time. In this problem, velocity is given as a function v(t) = 400 - 20t, which means the cyclist's speed changes linearly over time from 400 m/min to 200 m/min during the first 10 minutes.
Video consigliato:
Percorso guidato
10:17
Using The Velocity Function

Definite Integral for Displacement

The total distance traveled over a time interval can be found by integrating the velocity function over that interval. The definite integral of v(t) from t=0 to t=10 gives the net displacement, representing how far the cyclist has traveled in those 10 minutes.
Video consigliato:
Percorso guidato
05:43
Definition of the Definite Integral

Limits of Integration and Time Interval

The problem specifies the time interval 0 ≤ t ≤ 10 minutes. Setting the correct limits of integration ensures the calculation covers the entire duration of interest, capturing the cyclist's motion from start to 10 minutes exactly.
Video consigliato:
Percorso guidato
11:11
Improper Integrals: Infinite Intervals
Pratica correlata
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]?

88
views
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.

40
views
Domanda del libro di testo

Variable gravity At Earth’s surface, the acceleration due to gravity is approximately g=9.8 m/s² (with local variations). However, the acceleration decreases with distance from the surface according to Newton’s law of gravitation. At a distance of y meters from Earth’s surface, the acceleration is given by a(y) = - g / (1+y/R)², where R=6.4×10⁶ m is the radius of Earth.


b. Use the Chain Rule to show that dv/dt = 1/2 d/dy(v²).

69
views
Domanda del libro di testo

40–43. Population growth


Starting with an initial value of P(0)=55, the population of a prairie dog community grows at a rate of P′(t)=20−t/5 (prairie dogs/month), for 0≤t≤200, where t is measured in months.


b. Find the population P(t), for 0≤t≤200.

22
views
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?

62
views
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

50
views