Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problema 6.7.21

Capitolo 6, Problema 6.7.21

Work from force How much work is required to move an object from x=0 to x=3 (measured in meters) in the presence of a force (in N) given by F(x)=2x acting along the x-axis?

Guida verificata passo dopo passo

Identify the given force function: $F(x) = 2x$, which varies with position $x$ along the x-axis.

Recall that work done by a variable force along a straight line from $x=a$ to $x=b$ is given by the definite integral: $W = \int_{a}^{b} F(x) \, dx$.

Set up the integral for the work done moving the object from $x=0$ to $x=3$: $W = \int_{0}^{3} 2x \, dx$.

Integrate the function \$2x$ with respect to $x$: find the antiderivative of \$2x$, which is $x^2$.

Evaluate the definite integral by substituting the limits $x=3$ and $x=0$ into the antiderivative and subtracting: $W = [x^2]_{0}^{3} = 3^2 - 0^2$.

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Work Done by a Variable Force

Work done by a force that varies with position is calculated by integrating the force function over the displacement interval. Instead of using W = F × d, we use the integral W = ∫ F(x) dx from the initial to the final position to account for changes in force.

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05:40

Work Done On A Spring (Hooke's Law)

Definite Integral in Calculus

A definite integral computes the accumulation of quantities, such as area under a curve, over a specific interval. In this context, integrating F(x) from x=0 to x=3 sums the incremental work done at each point along the path.

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05:43

Definition of the Definite Integral

Force as a Function of Position

When force depends on position, it is expressed as F(x). Understanding how to interpret and manipulate such functions is essential for setting up the integral correctly and solving for work done over a distance.

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5:20

Relations and Functions

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Determine the area of the shaded region in the following figures.

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