Integration
Integral of a function (f(x)) over an interval ([a, b]) can be expressed as:
[Tex]\int_{a}^{b} f(x) , dx[/Tex]
This notation represents the area under the curve of the function f(x) between the points x = a and x = b. To understand this concept, consider dividing the interval [a, b] into smaller subintervals. Let’s say we have n subintervals, each of width \Delta x:
[Tex]\Delta x = \frac{b – a}{n}[/Tex]
Now, we approximate the area under the curve by summing up the areas of rectangles. For each subinterval i, we take the value of f(x) at some point xi within that subinterval and multiply it by the width (\Delta x):
(Area of rectangle)i = [Tex]f(x_i) \cdot \Delta x[/Tex]
Riemann sum is given by:
[Tex]R_n = \sum_{i=1}^{n} f(x_i) \cdot \Delta x[/Tex]
The integral is defined as the limit of this Riemann sum as (n) approaches infinity:
[Tex]\int_{a}^{b} f(x) dx = \lim_{{n \to \infty}} R_n[/Tex]
Calculus Cheat Sheet
Calculus is a branch of mathematics that studies the properties and behavior of functions, rates of change, limits, and infinite series. Calculus has many applications in science, engineering, economics, and other fields. However, calculus can also be challenging to learn and master, especially for beginners.
That is why we have prepared this calculus cheat sheet, a handy reference guide covering the most important concepts, formulas, rules, and calculus examples. Whether you need a quick review, a study aid, or a problem solver, this cheat sheet will help you ace calculus with ease.
Read: Calculus in Maths