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.R.7a
Capitolo 6, Problema 6.R.7a

An oscillator The acceleration of an object moving along a line is given by a(t) = 2 sin πt/4. The initial velocity and position are v(0)= −8/π and s(0)=0.
 a. Find the velocity and position for t≥0.

Guida verificata passo dopo passo
1
Start with the given acceleration function: \(a(t) = 2 \sin\left(\frac{\pi t}{4}\right)\). Recall that acceleration is the derivative of velocity, so \(a(t) = v'(t)\).
Integrate the acceleration function to find the velocity function: \(v(t) = \int a(t) \, dt = \int 2 \sin\left(\frac{\pi t}{4}\right) dt\). Remember to include the constant of integration \(C_1\).
Use the initial condition for velocity, \(v(0) = -\frac{8}{\pi}\), to solve for the constant \(C_1\) after integrating.
Next, integrate the velocity function to find the position function: \(s(t) = \int v(t) \, dt\). Again, include a constant of integration \(C_2\).
Use the initial position condition \(s(0) = 0\) to solve for the constant \(C_2\). This will give you the complete expressions for velocity \(v(t)\) and position \(s(t)\) for \(t \geq 0\).

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:
3m

Concetti chiave

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

Relationship Between Acceleration, Velocity, and Position

Acceleration is the derivative of velocity with respect to time, and velocity is the derivative of position. To find velocity and position from acceleration, you integrate acceleration once to get velocity and integrate velocity to get position, applying initial conditions to determine constants of integration.
Video consigliato:
Percorso guidato
06:15
Derivatives Applied To Acceleration

Integration of Trigonometric Functions

Integrating functions like sin(kt) involves using standard integral formulas, such as ∫sin(kt) dt = -cos(kt)/k + C. Understanding how to handle the coefficient inside the sine function is essential to correctly find antiderivatives when solving for velocity and position.
Video consigliato:
Percorso guidato
6:04
Introduction to Trigonometric Functions

Applying Initial Conditions

Initial conditions, such as v(0) and s(0), are used to find the constants of integration after integrating acceleration and velocity. These conditions ensure the solution matches the physical scenario and provide unique expressions for velocity and position.
Video consigliato:
Percorso guidato
05:03
Initial Value Problems
Pratica correlata
Domanda del libro di testo

Spring work


b. It takes 50 N of force to stretch a spring 0.2 m from its equilibrium position. How much work is needed to stretch it an additional 0.5 m?

58
views
Domanda del libro di testo

Two methods The region R in the first quadrant bounded by the parabola y = 4-x² and coordinate axes is revolved about the y-axis to produce a dome-shaped solid. Find the volume of the solid in the following ways:


b. Apply the shell method and integrate with respect to x.

59
views
Domanda del libro di testo

An area function Consider the functions y = x²/a and y = √x/a, where a>0. Find A(a), the area of the region between the curves.

47
views
Domanda del libro di testo

Pumping water A water tank has the shape of a box that is 2 m wide, 4 m long, and 6 m high.


b. If the water in the tank is 2 m deep, how much work is required to pump the water to a level of 1 m above the top of the tank?

47
views
Domanda del libro di testo

43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.


The region bounded by the graphs of y = 2x,y = 6−x, and y = 0 is revolved about the line y = −2 and the line x = −2. Find the volumes of the resulting solids. Which one is greater?

79
views
Domanda del libro di testo

Lifting problem A 4-kg mass is attached to the bottom of a 5-m, 15-kg chain. If the chain hangs from a platform, how much work is required to pull the chain and the mass onto the platform?

68
views