In physics, work done and gravitational potential energy are linked to integration in mathematics because both concepts involve accumulating small contributions over a distance.

1. Work Done: Work done by a force is the integral of the force over a distance. Mathematically, if a force varies with position , the total work done to move an object from point to is given by:

$$ W = \int_{x_1}^{x_2} F(x) \, dx $$

$$ W = mg \Delta h $$

1. Gravitational Potential Energy: Gravitational potential energy at a height is the energy stored due to an object’s position in a gravitational field. It is derived by integrating the gravitational force with respect to height:

$$ U = -\int F(h) \, dh = -\int mg \, dh = -mgh $$

Where:

• U is the gravitational potential energy
• m is the mass of the object
• g is the acceleration due to gravity
• h is the height above a reference point

Relationship through Integration:

The work-energy theorem states that the work done on an object is equal to the change in its energy. Since work done in lifting an object results in a change in gravitational potential energy, the integral for work directly gives the potential energy function:

$$ W = \Delta U = mgh $$

In essence, integration helps accumulate the small forces applied over a distance to give a complete picture of work done and the resulting change in gravitational potential energy.

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