Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problema 6.1.3a

Capitolo 6, Problema 6.1.3a

Given the velocity function of an object moving along a line, explain how definite integrals can be used to find the displacement of the object.

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Understand that the velocity function \(v(t)\) gives the rate of change of the object's position with respect to time, i.e., \(v(t) = \frac{ds}{dt}\), where \(s(t)\) is the position function.

Recall that displacement over a time interval \([a, b]\) is the change in position, which can be found by integrating the velocity function over that interval.

Set up the definite integral of the velocity function from \(a\) to \(b\): \(\int_{a}^{b} v(t) \, dt\).

Interpret this integral as the net area under the velocity curve between \(t = a\) and \(t = b\), where areas above the time-axis contribute positively and areas below contribute negatively to displacement.

Evaluate the definite integral to find the displacement, which represents how far and in what direction the object has moved from time \(a\) to time \(b\).

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Velocity Function

The velocity function describes the rate of change of an object's position with respect to time. It can be positive or negative, indicating direction along a line. Understanding velocity is essential because it directly relates to how the position changes over time.

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10:17

Using The Velocity Function

Definite Integral as Net Change

A definite integral of a velocity function over a time interval calculates the net change in position, or displacement. It sums the signed areas under the velocity curve, accounting for direction, thus providing the overall movement from start to end time.

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05:43

Definition of the Definite Integral

Displacement vs. Distance

Displacement is the net change in position, which can be positive, negative, or zero, depending on direction. It differs from total distance traveled, which sums all movement regardless of direction. Definite integrals of velocity yield displacement, not total distance.

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08:56

Using The Acceleration Function Example 1

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55–58. Marginal cost Consider the following marginal cost functions.

a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.

C′(x) = 300+10x−0.01x²

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6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.

Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.

a. What is the radius of a cylindrical shell at a point x in [0, 4]?

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Let R be the region in the first quadrant bounded above by the curve y=2−x² and bounded below by the line y=x. Suppose the shell method is used to determine the volume of the solid generated by revolving R about the y-axis.

b. What is the height of a cylindrical shell at a point x in [0, 2]?

138

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13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.

b. Find the displacement over the given interval.

v(t) = 50e^−2t on [0, 4]

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Emptying a water trough A water trough has a semicircular cross section with a radius of 0.25 m and a length of 3 m (see figure).

a. How much work is required to pump the water out of the trough (to the level of the top of the trough) when it is full?

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A vertical spring A 10-kg mass is attached to a spring that hangs vertically and is stretched 2 m from the equilibrium position of the spring. Assume a linear spring with F(x) = kx.

a. How much work is required to compress the spring and lift the mass 0.5 m?

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