Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.1.28b

Chapter 6, Problem 6.1.28b

Probe speed A data collection probe is dropped from a stationary balloon, and it falls with a velocity (in m/s) given by v(t) = 9.8t, neglecting air resistance. After 10 s, a chute deploys and the probe immediately slows to a constant speed of 10 m/s, which it maintains until it enters the ocean.

b. How far does the probe fall in the first 30 s after it is released?

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Identify the velocity function for the probe in two time intervals: from 0 to 10 seconds, the velocity is given by \(v(t) = 9.8t\), and from 10 to 30 seconds, the velocity is constant at \(v(t) = 10\) m/s.

To find the distance fallen in the first 10 seconds, integrate the velocity function \(v(t) = 9.8t\) with respect to time over the interval \([0, 10]\). This gives the displacement during free fall before the chute deploys: \(\int_0^{10} 9.8t \, dt\).

To find the distance fallen from 10 to 30 seconds, use the constant velocity \(v(t) = 10\) m/s. Since velocity is constant, the distance is velocity multiplied by time: \(10 \times (30 - 10)\).

Add the two distances obtained from the two intervals to get the total distance fallen in the first 30 seconds: \(\text{distance}_1 + \text{distance}_2\).

Express the total distance as the sum of the integral result and the constant velocity distance, which represents the full distance the probe falls in the first 30 seconds after release.

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Key Concepts

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

Velocity and Displacement Relationship

Velocity is the rate of change of displacement with respect to time. To find the distance fallen, you integrate the velocity function over the given time interval. This process accumulates the total displacement from the velocity data.

Piecewise Functions in Motion

When an object’s velocity changes behavior at a certain time, its motion is described by a piecewise function. Here, velocity changes at 10 seconds, requiring separate integration for each time segment to find total displacement.

Definite Integration for Distance Calculation

Definite integration of velocity over a time interval gives the exact displacement during that period. For constant velocity, displacement is velocity multiplied by time; for variable velocity, integration is necessary.

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Work done by a spring A spring on a horizontal surface can be stretched and held 0.5 m from its equilibrium position with a force of 50 N.

b. How much work is done in compressing the spring 0.5 m from its equilibrium position?

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In the design of solid objects (both artificial and natural), the ratio of the surface area to the volume of the object is important. Animals typically generate heat at a rate proportional to their volume and lose heat at a rate proportional to their surface area. Therefore, animals with a low SAV ratio tend to retain heat, whereas animals with a high SAV ratio (such as children and hummingbirds) lose heat relatively quickly.

b. What is the SAV ratio of a ball with radius a?

Textbook Question

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.

Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers.

b. What is the inner radius of a cross section of the solid at a point y in [1, 3]?

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Textbook Question

Region R is revolved about the line y=1 to form a solid of revolution.

c. Write an integral for the volume of the solid.

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.

c. The position at t=5

Textbook Question

Let R be the region bounded by the curve y=cos^−1x and the x-axis on [0, 1]. A solid of revolution is obtained by revolving R about the y-axis (see figures).

b. Find an expression for the area A(y) of a cross section of the solid at a point y in [0,π/2].