Goal: compute total amount of change over a period of time.
Walking speed: $\displaystyle \diff{x}{t} = 1$
Walking speed: $\displaystyle \diff{x}{t} = 1 + 3t$
Distance = rate $\times \Delta t$
Let $f(t)=1+3t$
$t_i = i\Delta t$, for $ i=1, 2, \ldots, n$.
Rate=`{}`
Distance: `{}`
`{}`
Define definite integral (Riemann integral) as limit as $n \to \infty$. ($\Delta t = 2/n$)
\begin{align*} \text{Amount walked} &= \int_0^2 f(t) dt\\ &= \lim_{n \to \infty} \sum_{i=1}^n f(t_{i-1})\Delta t\\ &= \lim_{n \to \infty} \sum_{i=1}^n f(t_{i})\Delta t \end{align*}The definite integral of the function $f(t)$ over the interval $t \in [a,b]$:
Example: estimate $\displaystyle \int_{-4}^{-1} x^2 dx$ with left and right Riemann sums of 6 intervals.
$a=-4$, $b=-1$, $\Delta x =(b-a)/6 = (-1-(-4))/6 = 3/6 = 1/2$
$x_0 = -4$, $x_1=-3.5$, $x_2=-3$, $x_3=-2.5$, $x_4=-2$, $x_5=-1.5$, $x_6=-1$
Intervals: $[-4,-3.5]$, $[-3.5, -3]$, $[-3, -2.5]$, $[-2.5, -2]$, $[-2, -1.5]$, $[-1.5,-1]$
$\displaystyle I_l = \sum_{i=1}^6 (x_{i-1})^2 \Delta x$ $= x_0^2\Delta x + x_1^2\Delta x + x_2^2\Delta x + x_3^2\Delta x + x_4^2\Delta x + x_5^2\Delta x$
$=(-4)^2 0.5+ (-3.5)^2 0.5 + (-3)^2 0.5 + (-2.5)^2 0.5 + (-2)^2 0.5 + (-1.5)^2 0.5$
$=24.875$
$\displaystyle I_r = \sum_{i=1}^6 (x_{i})^2 \Delta x = x_1^2\Delta x + x_2^2\Delta x + x_3^2\Delta x + x_4^2\Delta x + x_5^2\Delta x + x_6^2\Delta x$
$=(-3.5)^2 0.5 + (-3)^2 0.5 + (-2.5)^2 0.5 + (-2)^2 0.5 + (-1.5)^2 0.5 + (-1)^2 0.5$
$=17.375$