A cylindrical resistor is 5.0 mm in diameter and \(1.5 \mathrm{cm}\) long. It's made of a composite material whose resistivity varies from one end to the other according to the equation \(\rho=\rho_{0}(1+x / L) e^{x / L},\) for \(0 \leq x \leq L,\) where \(\rho_{0}=2.41 \times 10^{-3} \Omega \cdot \mathrm{m} .\) Find its resistance.

Electrical resistivity is a fundamental property of a material that quantifies how strongly it opposes the flow of electric current. A higher resistivity means the material is more resistant to the flow of electricity. This property is useful in understanding how well materials conduct electrical current.

Resistivity is typically denoted by the Greek letter \rho, and it is intrinsic to materials; for example, metals generally have low resistivities and are good conductors, while insulators like rubber have high resistivities. Resistivity is also dependent on temperature and can even change with the direction of current in anisotropic materials.

The most common use of resistivity in calculations is when determining the resistance of a material. Resistance, represented by the symbol R, can be thought of as the practical 'effect' of resistivity in a given object, and it is calculated using the formula: \[ R = \rho \frac{L}{A} \] where L is the length through which the current flows, A is the cross-sectional area of the material, and \rho is the electrical resistivity of the material.

Unlike resistance, which varies with the dimensions of the object, resistivity is a characteristic property of the material itself. It's important to understand this distinction when working on problems involving resistance calculation.

A cylindrical resistor is a practical component in electronics where resistance is needed in a circular cross-sectional shape. The calculation of resistance for a cylindrical resistor requires knowing not just resistivity but also the geometric dimensions of the cylinder.

In calculating the resistance of a cylindrical object, the cross-sectional area (A) is particularly important. For a cylinder, A is determined by the radius (r) of the base of the cylinder, with the formula \[ A = \pi r^2 \]. The radius must be measured or converted to meters, as the SI unit for area is square meters (m^2).

The length (L) of the cylinder is the distance over which the current travels, and in the context of a cylindrical resistor, it is the length of the material from one end of the cylinder to the other. It is also converted to meters when calculating resistance to maintain consistent units.

In the given exercise, it is crucial to note that the resistivity (\rho) is not constant and varies along the length of the resistor. Therefore, an integration approach is necessary for the calculation.

Integration is a mathematical tool that is extensively used in physics to solve problems involving continuous variables. Essentially, integration can be thought of as the process of calculating the area under a curve. This principle is applicable in various situations, such as calculating the work done by a force, the charge distributed over a surface, or, as in our problem, the resistance of an object with variable resistivity.

When resistivity varies along the length of an object, like our cylindrical resistor, we must 'add up' the infinitesimally small resistances along this length. The integration allows us to do this, turning a complicated variable-resistivity problem into a solvable integral. The general form for calculating resistance with a variable resistivity is: \[ R = \int{\rho \frac{dx}{A}} \] where \rho is a function of x, and A is a constant.

In the case of the cylindrical resistor, we are dealing with the integration of a function of x that defines resistivity. Integrating this function over the entire length of the resistor will yield the total resistance. This process accounts for the fact that at each sliver of the cylinder (each increment of x) the resistivity is slightly different, and all contributions are required for the final calculation of resistance.