Integrals and the Trapezoidal Rule

What are integrals?

Suppose the velocity of a car at any time $x$ is given by a function $f(x)$ and we would like to find the distance traveled by it in a certain time frame. Recall that the distance traveled by the car is represented as the area under velocity-time graph. Because $f(x)$ can be any complex function, we need integrals to compute the area under a graph.

The integral of a function $\int_a^b f(x)\cdot dx$ represents the area under the curve $f(x)$ between the points $a$ and $b$. For univariate functions, the integral represents the area under the curve, whereas, for multivariate functions, it represents the volume (3D space) and its generalization in higher dimensions. Mathematically, an integral is defined as follows:

An integral is an inverse operation of the derivative. In other words, the integral of a function's derivative results in the same function. Similarly, the derivative of an integral of a function is the same function.

As shown in the figure below, the integral can also be considered a summation of areas of infinite rectangles of the height $f(x)$ and the width $dx$.

Given below are the integrals of some of the important functions:

• Constant: $\int \text{c}_1 \cdot dx = \text{c}_1 x + \text{c}_2$, where $\text{c}_1$ and $\text{c}_2$ are constants.

• Polynomial: $\int x^n \cdot dx = \frac{x^{n+1}}{n+1} + \text{c}$ ...