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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 6.1.10a
Chapter 6, Problem 6.1.10a

9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.
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a. The displacement between t=0 and t=5

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Identify that displacement is the integral of velocity over time, which geometrically corresponds to the area under the velocity-time graph between the given time interval.
Divide the graph into simple geometric shapes between t = 0 and t = 5. Here, the graph consists of a rectangle from t = 0 to t = 3 with height 3, and a triangle from t = 3 to t = 5 where velocity decreases linearly from 3 to 0.
Calculate the area of the rectangle: Area = base \(\times\) height = (3 - 0) \(\times\) 3.
Calculate the area of the triangle: Area = \(\frac{1}{2}\) \(\times\) base \(\times\) height = \(\frac{1}{2}\) \(\times\) (5 - 3) \(\times\) 3.
Add the areas of the rectangle and triangle to find the total displacement between t = 0 and t = 5.

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Key Concepts

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

Velocity and Displacement

Velocity represents the rate of change of position with respect to time. Displacement is the net change in position over a time interval and can be found by integrating the velocity function over that interval. Positive velocity indicates movement in the positive direction, while negative velocity indicates movement in the opposite direction.
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Using The Velocity Function

Area Under the Velocity-Time Graph

The displacement over a time interval is equal to the area under the velocity-time graph between those times. For piecewise linear velocity graphs, this area can be calculated by summing the areas of geometric shapes like rectangles and triangles formed under the curve.
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Estimating the Area Under a Curve Using Left Endpoints

Piecewise Functions and Integration

When velocity changes in segments, the function is piecewise. To find displacement, integrate each segment separately and sum the results. This approach simplifies calculations, especially when velocity is constant or changes linearly within intervals.
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Piecewise Functions
Related Practice
Textbook Question

Oil production An oil refinery produces oil at a variable rate given by Q'(t) = <1x3 matrix>, where is measured in days and is measured in barrels. 


a. How many barrels are produced in the first 35 days?

Textbook Question

17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.


a. Determine the position function, for t≥0, using the antiderivative method


v(t) = 9−t² on [0, 4]; s(0)=−2

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Textbook Question

{Use of Tech} Oscillating motion A mass hanging from a spring is set in motion, and its ensuing velocity is given by v(t) = 2π cos πt, for t≥0. Assume the positive direction is upward and s(0)=0. 


a. Determine the position function, for t≥0.

Textbook Question

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


a. Find the area of region R₁.


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

A surface is generated by revolving the line f(x)=2−x, for 0≤x≤2, about the x-axis. Find the area of the resulting surface in the following ways.


a. Using calculus

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


a. The area of the region bounded by y=x and x=y^2 can be found only by integrating with respect to x.

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