Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 5 - Integration

Problema 5.4.35

Capitolo 5, Problema 5.4.35

Average velocity The velocity in m/s of an object moving along a line over the time interval [0,6] is v (t) = t² + 3t. Find the average velocity of the object over this time interval.

Guida verificata passo dopo passo

Step 1: Recall the formula for average velocity over a time interval [a, b]. It is given by:

\frac{v_{avg}}{= \frac{1_{b-a}}{\int_{a b}}}

Step 2: Substitute the given time interval [0, 6] into the formula. The average velocity becomes:

\frac{v_{avg}}{= \frac{1_{6-0}}{\int_{06}}}

Step 3: Write the integral expression for the velocity function v(t) = t² + 3t. The integral becomes:

\int_{06} (t²+3t) dt

Step 4: Compute the integral of the function t² + 3t. Use the power rule for integration:

\int t² dt = \frac{t³}{3}

and

\int 3t dt = \frac{3t²}{2}

. Combine these results to find the antiderivative.

Step 5: Evaluate the definite integral by substituting the limits of integration (0 and 6) into the antiderivative. Then divide the result by the length of the interval (6 - 0) to find the average velocity.

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Average Velocity

Average velocity is defined as the total displacement divided by the total time taken. In calculus, it can be calculated using the formula: average velocity = (s(b) - s(a)) / (b - a), where s(t) is the position function and [a, b] is the time interval. For the given problem, we need to integrate the velocity function over the interval and then divide by the length of the interval.

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Average Value of a Function

Velocity Function

The velocity function describes how the velocity of an object changes over time. In this case, the velocity is given by v(t) = t² + 3t, which is a polynomial function. Understanding this function is crucial for determining the object's behavior over the specified time interval, as it provides the rate of change of position with respect to time.

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Using The Velocity Function

Integration

Integration is a fundamental concept in calculus used to find the area under a curve, which in the context of motion, represents the total displacement. To find the average velocity, we need to integrate the velocity function v(t) over the interval [0, 6] and then divide the result by the length of the interval. This process allows us to calculate the total change in position over the given time period.

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Integration by Parts for Definite Integrals

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102

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