A variable force of 5 x^-2 newtons moves an object along a...

A variable force of $5 x^{-2}$ newtons moves an object along
a straight line when it is $x$ meters from the origin. Calculate
the work done in moving the object from $x=1 \mathrm{~m}$ to $x=10 \mathrm{~m}$.

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Key Concepts

Definite Integral

The definite integral is a fundamental concept in calculus used to calculate the accumulation of quantities, such as work, over a specific interval. When the force varies with position, the work done is computed by integrating the force function over the displacement interval.

Work

Work is a measure of energy transfer that occurs when a force is applied over a distance. In physics, work is calculated as the integral of the force along the path of motion, which sums the incremental contributions of the force over the displacement.

Variable Force

A variable force is one whose magnitude and/or direction changes with position. Unlike a constant force, calculating work done by a variable force requires integration because the force is not uniform over the path of motion.

Integration of Power Functions

When the force is expressed as a power function of position (such as an inverse square function), techniques for integrating power functions are applied. This involves finding the antiderivative of the function, which is then used to evaluate the work done over the specified limits.

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