# Integral Calculus ## Average Values :::{.proposition title="Integral formula for average value"} \[ \mu_f = \frac{1}{b-a}\int_a^b f(t) dt \] ::: :::{.proof title="?"} Apply MVT to \(F(x)\). ::: ## Area Between Curves Area in polar coordinates: $$ A = \int_{r_1}^{r_2} \frac{1}{2}r^2(\theta) ~d\theta $$ ## Solids of Revolution \[ \text{Disks} && A = \int \pi r(t)^2 ~dt \\ \text{Cylinders} && A = \int 2\pi r(t)h(t) ~dt .\] ## Arc Lengths \[ L &= \int ~ds && ds = \sqrt{dx^2 + dy^2} \\ &= \int_{x_0}^{x_1}\sqrt{1 + \dd{y}{x}}~dx \\ &= \int_{y_0}^{y_1}\sqrt{\dd{x}{y} + 1}~dy \] \[ SA = \int 2 \pi r(x) ~ds \] ## Center of Mass Given a density $\rho(\mathbf x)$ of an object $R$, the $x_i$ coordinate is given by \[ x_i = \frac {\displaystyle\int_R x_i\rho(x) ~dx} {\displaystyle\int_R \rho(x)~dx} \] ## Big List of Integration Techniques Given $f(x)$, we want to find an antiderivative $F(x) = \int f$ satisfying $\frac{\partial}{\partial x}F(x) = f(x)$ - Guess and check: look for a function that differentiates to \(f\). - $u\dash$ substitution - More generally, any change of variables \[ x = g(u) \implies \int_a^b f(x)~dx = \int_{g^{-1}(a)}^{g^{-1}(b)} (f\circ g)(x) ~g'(x)~dx \] ### Integration by Parts: The standard form: \[ \int u dv = uv - \int v du \] - A more general form for repeated applications: let $v^{-1} = \int v$, $v^{-2} = \int\int v$, etc. \[ \int_a^b uv &= uv^{-1}\bigg\rvert_a^b - \int_a^b u^{1} v^{-1}\\ &= uv^{-1} - u^1v^{-2}\bigg\rvert_a^b + \int_a^b u^2v^{-2} \\ &= uv^{-1} - u^1v^{-2} + u^2v^{-3}\bigg\rvert_a^b - \int_a^b u^3v^{-3} \\ &\quad\vdots \\ \implies \int_a^b uv &= \sum_{k=1}^n (-1)^k u^{k-1}v^{-k} \bigg\rvert_a^b + (-1)^n\int_a^b u^nv^{-n} \] - Generally useful when one term's \(n\)th derivative is a constant. ### Shoelace Method - Note: you can choose \(u\) or \(v\) equal to 1! Useful if you know the derivative of the integrand. Derivatives | Integrals | Signs | Result --- | --- | -- | --- | --- $u$ | $v$ | NA | NA $u'$ | $\int v$ | $+$ | $u\int v$ $u''$ | $\int\int v$ | $-$ | $-u'\int\int v$ $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ Fill out until one column is zero (alternate signs). Get the result column by multiplying diagonally, then sum down the column. ### Differentiating under the integral \[ \frac{\partial}{\partial x} \int_{a(x)}^{b(x)} f(x, t) dt - \int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(x, t) dt &= f(x, \cdot)\frac{\partial}{\partial x}(\cdot) \bigg\rvert_{a(x)}^{b(x)} \\ &= f(x, b(x))~b'(x) - f(x, a(x))~a'(x) \] :::{.proof title="?"} Let \(F(x)\) be an antiderivative and compute \(F'(x)\) using the chain rule. \todo[inline]{ For constants, this should allow differentiating under the integral when \(f, f_x\) are "jointly continuous" } - LIPET: Log, Inverse trig, Polynomial, Exponential, Trig: generally let \(u\) be whichever one comes first. - The ridiculous trig sub: for any integrand containing only trig terms - Transforms *any* such integrand into a rational function of \(x\) - Let $u = 2\tan^{-1}x, ~du = \frac{2}{x^2+1}$, then \[ \int_a^b f(x)~dx = \int_{\tan\frac{a}{2}}^{\tan\frac{b}{2}} f(u)~du \] ::: :::{.example title="?"} \[ \int_0^{\pi/2} \frac{1}{\sin \theta}~d\theta = 1/2 \] ::: - Trigonometric Substitution \[ \sqrt{a^2-x^2} && \Rightarrow && x = a\sin(\theta) &&dx = a\cos(\theta)~d\theta \\ \sqrt{a^2+x^2} && \Rightarrow && x = a\tan(\theta) &&dx = a\sec^2(\theta)~d\theta \\ \sqrt{x^2 - a^2} && \Rightarrow && x = a \sec(\theta) &&dx = a\sec(\theta)\tan(\theta)~d\theta \] ### Partial Fractions ### Trigonometric Substitution \todo[inline]{Completing the square} - Trig Formulas \[ \sin^2(x) && = && \frac{1}{2}(1-2\cos x) \\ && = && \\ && = && \\ && = && \\ && = && \\ \] \todo[inline]{ Trig functions, double angle formulas. } - Products of trig functions - Setup: $\int \sin^a(x) \cos^b(x) ~dx$ - Both \(a,b\) even: $\sin(x)\cos(x) = \frac{1}{2} \sin(x)$ - \(a\) odd: $\sin^2 = 1-\cos^2,~u=\cos(x)$ - \(b\) odd: $\cos^2 = 1-\sin^2,~u=\sin(x)$ - Setup: $\int \tan^a(x) \sec^b(x) ~dx$ - \(a\) odd: $\tan^2 = \sec^2 - 1,~ u = \sec(x)$ - \(b\) even: $\sec^2 = \tan^2 - 1, u = \tan(x)$ Other small but useful facts: \[ \int_0^{2\pi} \sin \theta~d\theta = \int_0^{2\pi} \cos \theta~d\theta = 0 .\] ## Optimization - Critical points: boundary points and wherever \(f'(x) = 0\) - Second derivative test: - $f''(p) > 0 \implies p$ is a min - $f''(p) < 0 \implies p$ is a max - Inflection points of \(h\) occur where the _tangent_ of \(h'\) changes sign. (Note that this is where \(h'\) itself changes sign.) - Inverse function theorem: The slope of the inverse is reciprocal of the original slope - If two equations are equal at exactly one real point, they are tangent to each other there - therefore their derivatives are equal. Find the \(x\) that satisfies this; it can be used in the original equation. - Fundamental theorem of Calculus: If \[ \int f(x) dx = F(b) - F(a) \implies F'(x) = f(x) .\] - Min/maxing - either derivatives of Lagranage multipliers! - Distance from origin to plane: equation of a plane \[ P: ax+by+cz=d .\] - You can always just read off the normal vector $\vector{n} = (a,b,c)$. So we have $\mathbf{n}\mathbf{x} = d$. - Since $\lambda \mathbf{n}$ is normal to $P$ for all $\lambda$, solve $\mathbf{n}\lambda \mathbf{n} = d$, which is $\lambda = \frac{d}{ \norm{\vector n}^2}$ - A plane can be constructed from a point \(p\) and a normal \(n\) by the equation \(np = 0\). - In a sine wave $f(x) = \sin(\omega x)$, the period is given by $2\pi/\omega$. If $\omega > 1$, then the wave makes exactly $\omega$ full oscillations in the interval $[0, 2\pi]$. - The directional derivative is the gradient dotted against a _unit vector_ in the direction of interest - Related rates problems can often be solved via implicit differentiation of some constraint function - The second derivative of a parametric equation is not exactly what you'd intuitively think! - For the love of god, remember the FTC! \[ \frac{\partial}{\partial x} \int_0^x f(y) dy = f(x) \] - Technique for asymptotic inequalities: WTS \(f < g\), so show \(f(x_0) < g(x_0)\) at a point and then show $\forall x > x_0, f'(x) < g'(x)$. Good for big-O style problems too. - Inflection points of \(h\) occur where the _tangent_ of \(h'\) changes sign. (Note that this is where \(h'\) itself changes sign.) - Inverse function theorem: The slope of the inverse is reciprocal of the original slope - If two equations are equal at exactly one real point, they are tangent to each other there - therefore their derivatives are equal. Find the \(x\) that satisfies this; it can be used in the original equation. - Fundamental theorem of Calculus: If \[ \int f(x) dx = F(b) - F(a) \implies F'(x) = f(x) .\] - Min/maxing - either derivatives of Lagranage multipliers! - Distance from origin to plane: equation of a plane \[ P: ax+by+cz=d .\]