Integral

${\int }_a^{b}f(x)dx=\underset{n\to \mathrm{\infty }}{lim}\sum _{i=0}^{n}f(x_i^{\ast })\mathrm{\Delta }x$

• Where:
  • $\mathrm{\Delta }x=\frac{b-a}{n}$
  • $x_i^{\ast }\in [x_{i-1},x_i]$
• If the limit exists, then $f(x)$ is integrable on the interval $[a,b]$
• The width of each bar $\mathrm{\Delta }x$ doesn't have to be the same
• Look it's the Riemann Sum
• $f(x)$ is the integrand, $a$ is the lower limit, $b$ is the upper limit
• The definite integral does not depend on $x$
  • $x$ is a dummy variable, you can change it with anything and the integral still evaluates to the same number
• You can think of the integral as the area under the curve between $[a,b]$
  • If the function goes negative, the area above the curve is negative
• If $f(x)$ is continuous on $[a,b]$, then $f(x)$ is integrable on $[a,b]$
  • Note that this isn't an iff, since jump discontinuities are still integrable
• Kinda the opposite of Derivative
  • Antiderivative
• Improper Integral

Properties

• For a constant $k\in \mathbb{R}$, ${\int }_a^{b}kdx=k(b-a)$
• Linearity:
  • ${\int }_a^b(f(x)\pm g(x))dx={\int }_a^{b}f(x)dx\pm {\int }_a^{b}g(x)dx$
  • ${\int }_a^{b}kf(x)dx=k{\int }_a^{b}f(x)dx$ for any $k\in \mathbb{R}$
• You can split up the integral: ${\int }_a^{b}f(x)dx={\int }_a^{c}f(x)dx+{\int }_c^{b}f(x)dx$
  • Note that this works for ANY $c\in \mathbb{R}$, even if $c$ is not between $a,b$, because of the next property
• ${\int }_a^{b}f(x)dx=-{\int }_b^{a}f(x)dx$
• ${\int }_a^{a}f(x)dx=0$
• If $f(x)\ge 0$ for $a\le x\le b$, then ${\int }_a^{b}f(x)dx\ge 0$
• If $f(x)\ge g(x)$ for $a\le x\le b$, then ${\int }_a^{b}f(x)dx\ge {\int }_a^{b}g(x)dx$
  • This means you can integrate both sides of an inequality
• If $m$ is the local minima and $M$ is the local maxima of $f(x)$ between $x\in [a,b]$, then $m(b-a)\le {\int }_a^{b}f(x)dx\le M(b-a)$
  • This is kinda like squeeze theorem for integrals
• $|{\int }_a^{b}f(x)dx|\le {\int }_a^b|f(x)|dx$
  • This is kinda like triangle inequality for integrals
  • This is because ${\int }_a^{b}f(x)dx$ may have some negative and some positive areas
• If $f(x)$ is an odd function, then ${\int }_{-a}^{a}f(x)dx=0$
• If $f(x)$ is an even function, then ${\int }_{-a}^{a}f(x)dx=2{\int }_0^{a}f(x)dx$

Area Between Two Curves

• Find the points of intersection
• Subtract the functions and take the definite integral between those points
  • It doesn't actually matter which curve your subtract, since

Double Integrals

• In math119, we only deal with smooth functions and integrals with closed bounds
• With multiple variables, the $+C$ actually becomes a function of the unused variables
• $\int {\int }_{Dxy}h(x,y)dxdy$
• This is like getting the volume under some function with 2 variables inside some region
  • If that function is $f(x,y)=1$, then you just get the area inside the region
  • The intuition for this is to think about the sums along one variable, which gives a function of the other variable, which you then sum again
• Make sure the bounds of the inner integral are based on the outer integral, if the bounds of the inner integral depend on the outer variable
  • If the inner bounds change when you do your sum, you need to account for that
    • Think about nested for loops
  • eg. if your bounds are $y=x^2,y=x$, and you do $\int \int h(x,y)dxdy$, then your inner bounds must be $\sqrt{y},y$
  • It doesn't matter which way you do first tho, but sometimes on direction is easier
    • You might be able to avoid computing an antiderivative until you have a substitution
    • If the function takes a single variable, integrate wrt the other variable first, so it's just a constant
• When you do the inner integral, you get an expression for the area of some slice at $x$
• If the area is a rectangle or cube (the bounds do not depend on any of the variables), and you have a product of functions of the variables, you can express it as the product of integrals instead (since you keep moving out "constants")
  • ${\int }_a^b{\int }_c^{d}xydydx={\int }_a^{b}x{\int }_c^{d}y\mathrm{d}y\mathrm{d}x={\int }_a^{b}x\mathrm{d}x{\int }_c^{d}y\mathrm{d}y$
• Normal integration properties still apply here