A line integral is a type of integral that calculates the total value of a function along a given curve or path. It takes into account both the function and the path, and is often used in physics and engineering to calculate work, flux, and other physical quantities.
The formula for computing a line integral is ∫C f(x,y) ds, where C is the curve or path, f(x,y) is the function being integrated, and ds is an infinitesimal element of length along the curve.
To compute a line integral numerically, you can divide the curve into smaller segments and approximate the integral using the trapezoidal rule or Simpson's rule. Alternatively, you can use software programs such as Mathematica or MATLAB to calculate the integral.
The function 4xz + 2y represents the value of the function along the given curve or path. In this case, it is the function that is being integrated, and its value will change depending on the values of x and y at each point along the curve.
Line integrals have many real-world applications, such as calculating the work done by a force along a curved path, finding the flux of a vector field through a closed surface, and determining the mass or charge of an object by integrating its density or charge distribution along a path.