= Integral
{title2=$\int$}
{wiki}
= Integration
{synonym}
An integral adds up contributions of a <function> across a <domain of integration>. For a <continuous function> $f$ on $[a,b]$, the <Riemann integral> is the common <limit> of its <Riemann sums> as the largest subinterval width tends to zero:
$$
\int_a^b f(x)\,dx
=\lim_{\max_i\Delta x_i\to0}\sum_i f(\xi_i)\Delta x_i,
\qquad \xi_i\in[x_{i-1},x_i],\quad \Delta x_i=x_i-x_{i-1}.
$$
Each product combines a value with the width over which it acts. When $f$ is nonnegative, the result is the <area> beneath its graph; values below the horizontal axis contribute with the opposite sign. The <signed area under a graph> can therefore vanish even when the graph encloses nonzero geometric <area>.
For a <velocity> $v(t)=1+2t$ on $0\le t\le3$, use $n$ equal time steps. The left and right <Riemann sums> are
$$
L_n=12-\frac9n,\qquad R_n=12+\frac9n.
$$
Because the <function> is increasing, they bound the <integral>. With six steps they give $10.5$ and $13.5$; with sixty steps they give $11.85$ and $12.15$. Both approach the same total, $12$. This is accumulation made exact by refining finite measurements.
\Image[/integral-riemann-sums.png]
{title=Left and right Riemann sums bracket the accumulated displacement of a particle with velocity 1 + 2t}
An <integral> retains units: <velocity> times time gives <displacement>, and a <probability density function> integrated over an <interval> gives a <probability>. Weighting a <density> by a <function> gives its <expectation>. The same operation serves different subjects because it expresses a common pattern, \b[value multiplied by extent, then accumulated].
The <fundamental theorem of calculus> often evaluates a <definite integral> by finding an <antiderivative>, but a finite sum or a numerical approximation is still useful when no convenient <antiderivative> is available. An <indefinite integral> denotes a family of <antiderivatives>; a <definite integral> denotes an accumulated value between specified endpoints. More general constructions, including <Lebesgue integration>, extend the class of <functions> and <domains> that can be handled.
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