# Integral ![[Integration#^sj74oi]] **The integral represents the area bounded by a [[Function (in Algebra)|function]] and the $x$-axis (between two specific $x$ values).** The area can be estimated by dividing the range into small rectangles and summing their areas[^1]. The exact area is then determined by calculating the [[Limit]] as the number of rectangles goes to infinity (and their individual widths go to 0). [^1]: [[Riemann Sum]] Computing integrals is useful because ![[The Essence of Calculus#^5dpw3u]] For examples see [[Calculating Function Averages]], [[Determining the Length of a Section of a Curve]], [[Calculating The Surface Area of a Smooth Object Defined by a Function Using Calculus]], and [[Calculating The Volume of a Smooth Object Defined by a Function Using Calculus]]. ^ia5q62 Per [[The Fundamental Theorem of Calculus]], the area below $f(x)$ between a and b is equal to: ![[The Fundamental Theorem of Calculus#^xr2kow]] Because computing integrals involves determining the function's [[Antiderivative]], and because the notation for the two is similar, we've grown to call antiderivatives *indefinite integrals*, with integrals themselves getting known as *definite integrals*. Integrals are often referred to as *definite integrals* because they refer to a particular area under a graph. On the other hand, the antiderivative is often referred to as the *indefinite integral*. ![[Antiderivative#^sm5a2j]] ## Integral Notation The following is an integral notation that indicates the area below $f$ between $a$ and $b$ where $a < b$. $F$ represents the antiderivative of $f$. $\int \limits_a^b f(x) dx = F\vert_a^b = F(b) - F(a)$ ## Integral Rules $ \begin{gather} \int \limits_a^\infty f(x) dx = \lim_{b \to \infty} \int \limits_a^b f(x) dx \\ \int \limits_{-\infty}^b f(x) dx = \lim_{a \to -\infty} \int \limits_a^b f(x) dx \\ \int \limits_{-\infty}^\infty f(x) dx = \lim_{a \to -\infty} \left[ \lim_{b \to \infty} \left( \int \limits_a^b f(x) dx \right) \right] \end{gather} $