![[Approximating Area Subinterval Gif.gif|300]] # Integration **Integrals** combine the concepts of [[limits]] and [[Sums and Series|summations]] to approximate the areas and volumes associated with a given function. *Integral calculus* is a branch of [[calculus]] that deals with the theory and application of integrals. > [!example] Applications of Integration > - [[Approximating Volume & Area Using Integration]] > - [[Average Value of Functions]] ## What Are Integrals? A derivative is to differentiation as an integral is to integration. When we perform integration on a function $f(x)$, we are *finding it’s integral*. The definitions of *indefinite* and *definite integrals* may seem disconnected at first glance (one is a reverse derivative while the other is related to calculating areas). They are united through the **Fundamental Theorem of Calculus**. ### The Indefinite Integral > [!summary] **Antiderivatives** > A function $F$ is an *antiderivative* of $f$ on an interval $I$ if: > $F'(x)=f(x)$ > for all $x$ in $I$. > --- > **General Form of an Antiderivative** > Let $F$ be an antiderivative of $f$ over an interval $I$. Then, > 1. for > Basically, an antiderivative is the original function of a derivative before it was differentiated. Using [[Kinematics & Dynamics|kinematics]] as an example: - The antiderivative of the velocity function, $v(x)$, would be the position function, $p(x)$. Meanwhile, finding the derivative of $p(x)$ would bring you back to the velocity function, $v(x)$. --- An **indefinite integral** can be thought of as the “formal notation” used to represent antiderivatives. > [!summary] **Definition of Indefinite Integrals** > The *collection of all antiderivatives* of $f$ is called the **indefinite integral** of $f$ with respect to $x$, and is denoted by: > > $\int f(x)dx$ > > - The symbol $\int$ is an integral sign. > - The function $f$ is the integrand of the integral. > - $x$ is the variable of integration (arbitrary/a dummy variable). The process of finding the antiderivative of a given function is called antidifferentiation or **integration** (title drop 😱). - Similarly to the differentiation rules used to find derivatives, there are several integration formulas that can assist in evaluating integrals. ### The Definite Integral The **definite integral** is the *limit of a riemann sum* as the number of subintervals approaches infinity. > [!abstract] **Definition of Definite Integrals** > If $f(x)$ is a function defined on an interval $[a,b]$, the **definite integral** of $f$, provided the limit exists, from $a$ to $b$ is given by > > $\int^{b}_{a} f(x)dx = \lim_{n \to \infty} \sum^{n}_{k=1} f(c_k) \Delta x$ > > If this limit exists the function $f(x)$ is said to be *integrable* on $[a,b]$ > It is also considered an *integrable function* > - $a$ represents the lower bound > - $b$ represents the upper bound ### The Fundamental Theorem of Calculus (FTOC) > [!summary] **The Fundamental Theorem of Calculus, Part I** > If $f(x)$ is continuous over an interval $[a,b]$, and the function $F(x)$ is defined by: > $g(x)= \int^{x}_{a} f(t)dt$ > then $g’(x)=f(x)$. > --- > Alternatively, this can be represented as: > $\frac{d}{dx}[\int^{x}_{a} f(t)dt]=f(x)$ > In other words: - The derivative of a definite integral with respect to its upper limit is the integrand evaluated at the upper limit. - Shows that, in some sense, integration is the opposite of differentiation. - This theorem *guarantees* that any integrable function has an antiderivative --- > [!summary] **The Fundamental Theorem of Calculus, Part II** > > If $f(x)$ is continuous over an interval $[a,b]$, then > $\int^{x}_{a} f(x)dx = F(b) - F(a)$ > where $F$ is any antiderivative of $f$, that is, a function such that $F’(x) = f(x)$. > --- > > A common notation for $F(b)-F(a)$ is $F(x)\Big|_a^b$ ## Integration Techniques | Technique | Description | | ------------------------------------------------- | ----------- | | [[Integration Using U-Substitution]] | | | [[Integration By Parts]] | | | [[Integration by Partial Fraction Decomposition]] | | | [[Integration Using Trigonometric Functions]] | | As seen with the more basic integration formulas, many [[Differentiation|derivatives]] have an “inverse” version The same is true for the chain rule and product rule of differentiation It’s somewhat inevitable that you will have to memorize certain integration formulas - It’s important to distinguish the important ones from others which can easily be derived as needed. **santi notes:** - integration is a linear operator - This is why we can “take the integral of both sides” and maintain the equality of an equation (similarly to how we can add identical numbers to both sides) - can’t easily take the limit of integrals - differentials are a different class than algebraic objects - slightly “weaker” - infinitely small next number that follows after $x$