>[!warning] This note is from high school, too lazy to write it again. ## Definition An integral / [[Anti-Derivative]] is the inverse of a [[Derivative]]. $\huge f(x) = \int_0^x\frac{\mathrm{d}f}{\mathrm{d}x}(x)\mathrm{d}x$ The visual way to define an is the [[Area]] / [[Measure]] under the curve of a [[Function]], for example, for the function of $f(x)=2x$, the integral of the function (through the [[Methods/Power Rule]]) is $x^2$. This means that the area enclosed by $f(x)$ from $0$ to $x$, the area is $x^2$. In the case of $2x$, the function is simply a right triangle, with the base of length $x$ and height of $2x$, hence $\frac{1}{2}(2x\cdot x)= x^2$. This relationship is also the integral $x^2$ is the *antiderivative* of $2x$. ### Finding the [[Area]] underneath a curve In the case of a constant or [[Linear Transformation|Linear]] function (such as $x^2$), finding the [[Area]] is simple because the [[Function]] give us straight line and simple rectangles and triangles with predefined area functions. For finding an area of a curved function, for an approximation we could use [[Finite]] rectangles spread out evenly of the function with width $\mathrm{d}x$, and add the area of each rectangle. $ \int_{a}^{b}f(x)\mathrm{d}x \approx \sum\limits_{x=a}^{b}f(x)\mathrm{d}x \\ $ While this is an approximation, for finer rectangles covering the function (eg. smaller $\mathrm{d}x$), this approaches the true area. $ \int_{a}^{b}f(x)\mathrm{d}x = \lim_{N \to \infty} \frac{1}{N}\sum\limits_{x=0}^{N}f\left(a+\frac{x}{N}(b - a)\right) \\ $ ![[../../../00 Asset Bank/Pasted image 20230706124045.png]] While in pure math we find the exact integral, it is also possible to [[Approximation|approximate]] the integral