Integration Symbol Calculator

The symbol $\int$ is an elongated S, chosen by Leibniz for summa — a sum. Every integral is the limit of a Riemann sum:

$\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i)\,\Delta x$

Reading a definite integral left to right:

$\int_{\underbrace{a}_{\text{lower limit}}}^{\overbrace{b}^{\text{upper limit}}} \underbrace{f(x)}_{\text{integrand}} \underbrace{dx}_{\text{variable}}$

• Limits $a$ and $b$ are the start and end of the interval. Without them it is an indefinite integral and the answer is a family of functions, so you must add $+C$.
• $dx$ names the variable of integration and closes the expression. It is not decoration: $\int x\,dt$ and $\int x\,dx$ are different problems.

Relatives: $\iint$ and $\iiint$ integrate over a region or solid, $\oint$ integrates around a closed curve, and $\int_a^\infty$ marks an improper integral that must be evaluated as a limit.

The Fundamental Theorem of Calculus turns the symbol into arithmetic. If $F' = f$ and $f$ is continuous on $[a,b]$:

$\int_a^b f(x)\,dx = \Big[F(x)\Big]_a^b = F(b) - F(a)$

The square bracket with sub- and superscript is the evaluation bar — it means "substitute the top, subtract the bottom".

What the result actually is. A definite integral is signed area: regions below the axis count negative. $\int_0^{2\pi}\sin x\,dx = 0$ even though the curve encloses 4 units of area.

Orientation. $\int_b^a f = -\int_a^b f$, and $\int_a^a f = 0$.

When no antiderivative exists — $e^{-x^2}$ has none in elementary terms — a calculator switches to numerical integration. Simpson's rule with an even $n$ and $h = (b-a)/n$:

$\int_a^b f \approx \frac{h}{3}\Big[f_0 + 4f_1 + 2f_2 + \cdots + 4f_{n-1} + f_n\Big]$

It assumes $f$ is smooth on $[a,b]$; a jump or a vertical asymptote inside the interval makes the estimate meaningless.

• Dropping $dx$. It marks where the integrand ends, which matters the moment you integrate a sum or substitute.
• Forgetting $+C$ on an indefinite integral, or writing $+C$ on a definite one, where it cancels.
• Subtracting in the wrong order. The evaluation bar is $F(b) - F(a)$, top minus bottom.
• Calling the answer "the area". It is signed area. For total area, split at each root and integrate $|f|$.
• Substituting without changing the limits. With $u = g(x)$ the limits become $g(a)$ and $g(b)$ — or convert back to $x$ before evaluating.
• Running a numerical rule across a discontinuity. Simpson's and the trapezoid rule assume continuity; split the interval at the break instead.