Skip to main content
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.R.7a
Chapter 6, Problem 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.

Verified step by step guidance
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 \(t \geq 0\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

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

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.
Recommended video:
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.
Recommended video:
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.
Recommended video:
05:03
Initial Value Problems
Related Practice
Textbook Question

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?

1
views
Textbook Question

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.

Textbook Question

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.

Textbook Question

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?

1
views
Textbook Question

58–61. Arc length Find the length of the following curves.

y = x³/6 + 1/2x on [1,2]

Textbook Question

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?

1
views