Key Properties of Integrals and Differentials Explained

Properties of Integrals

Integrals obey several important properties.

Sum Rule

The integral of the sum of two functions equals the sum of their integrals, and a constant factor can be taken outside the integral. For three points a, b, c, the integral from a to c can be split into the integral from a to b plus the integral from b to c.

This property is illustrated below:

Similarly, the integral from a to b is the negative of the integral from b to a.

Finally, if a function is non‑positive, its integral equals the negative of the area between the function and the x‑axis.

If the positive and negative areas are equal, the integral can be zero, as shown in the following figure:

Using Limits to Compute Integrals

Based on the definition of the integral, we can approximate the area under a simple function by summing the areas of narrow rectangles. Each rectangle’s width is Δx, and the left‑most coordinate of the i‑th rectangle is x_i. By rewriting the integral as a limit of sums and applying known formulas for sums of squares, we can evaluate the integral as the limit when the number of rectangles approaches infinity.

Differentials

Given a function f(x), we can define a finite difference Δf. The differential df represents an infinitesimal change in the value of f.

More precisely, a differential is an object that, when integrated, yields the original function, analogous to how a finite difference approximates the change in the function.

Properties of Differentials

Scalar Multiplication

If c is a constant and f is a function, then d(c·f) = c·df.

Addition

The differential of a sum equals the sum of the differentials: d(f+g) = df + dg.

Product

The differential of a product follows the product rule: d(f·g) = f·dg + g·df.

Chain Rule

If y = f(u) and u = g(x), then dy = f'(u)·du = f'(g(x))·g'(x)·dx.

Some Integral Properties Related to Differentials

Scalar Multiplication

The integral of a constant times a function equals the constant times the integral of the function.

Addition

The integral of a sum equals the sum of the integrals.

Splitting Domain of Integration

An integral over a domain can be split into integrals over subdomains, mirroring the additive property of differentials.