Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.1.7d

Chapter 6, Problem 6.1.7d

Displacement and distance from velocity Consider the graph shown in the figure, which gives the velocity of an object moving along a line. Assume time is measured in hours and distance is measured in miles. The areas of three regions bounded by the velocity curve and the t-axis are also given.

d. What is the displacement of the object over the interval [0,5]?

Verified step by step guidance

Understand that displacement over the interval [0,5] is the net change in position, which can be found by calculating the definite integral of the velocity function over that interval.

Recognize that the graph shows velocity values above and below the t-axis, indicating positive and negative velocities respectively. The areas given represent the absolute values of the integrals over subintervals.

Identify the three regions and their corresponding areas: from t=0 to t=1 (area = 12, velocity positive), from t=1 to t=3 (area = 16, velocity negative), and from t=3 to t=5 (area = 10, velocity positive).

Calculate the displacement by summing the signed areas: add the positive areas and subtract the negative area, i.e., displacement = 12 - 16 + 10.

Express the displacement as the sum of these signed areas, which represents the integral of velocity over [0,5], giving the net change in position (displacement).

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Key Concepts

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

Displacement as the Integral of Velocity

Displacement over a time interval is the net change in position, calculated as the definite integral of velocity with respect to time. It accounts for direction, so areas above the time-axis add positively, while areas below subtract, reflecting movement in opposite directions.

Interpreting Areas Under the Velocity Curve

The area between the velocity curve and the time-axis represents the distance traveled during that time segment. Positive velocity areas indicate forward movement, while negative velocity areas indicate backward movement. Summing these signed areas gives the net displacement.

Difference Between Displacement and Distance

Displacement is a vector quantity reflecting net change in position, considering direction, while distance is a scalar representing total ground covered regardless of direction. In velocity graphs, displacement is the net signed area, whereas distance is the sum of absolute areas.

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Textbook Question

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.

d. A piecewise function for s(t)

Textbook Question

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

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Textbook Question

Displacement and distance from velocity Consider the velocity function shown below of an object moving along a line. Assume time is measured in seconds and distance is measured in meters. The areas of four regions bounded by the velocity curve and the t-axis are also given.

d. What is the displacement of the object over the interval [0, 8]?

Textbook Question

Where do they meet? Kelly started at noon (t=0) riding a bike from Niwot to Berthoud, a distance of 20 km, with velocity v(t) = 15 / (t + 1)² (decreasing because of fatigue). Sandy started at noon (t=0) riding a bike in the opposite direction from Berthoud to Niwot with velocity u(t) = 20 / (t + 1)² (also decreasing because of fatigue). Assume distance is measured in kilometers and time is measured in hours.

d. More generally, if the riders’ speeds are v(t)=A(t+1)² and u(t)=B(t+1)² and the distance between the towns is D, what conditions on A, B, and D must be met to ensure that the riders will pass each other?

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Textbook Question

Compressing and stretching a spring Suppose a force of 30 N is required to stretch and hold a spring 0.2 m from its equilibrium position.

d. How much additional work is required to stretch the spring 0.2m if it has already been stretched 0.2m from its equilibrium position?

Textbook Question

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

d. Let f(x)=12x^2.. The area of the surface generated when the graph of f on [−4, 4] is revolved about the y-axis is twice the area of the surface generated when the graph of f on [0, 4] is revolved about the y-axis.