Integration: Adding Up Slivers

Slice an area into thin bars, add them all, and let the slices vanish.

16 minUpdated 21 October 2026Riemann sumdefinite integralantiderivativefundamental theorem

Lesson at a glanceShow the key points

∫ₐᵇ f(x) dx is the limit of the sum Σ f(xᵢ)Δx as Δx → 0: the area under the curve between a and b.
If dF/dx = f(x), then ∫ₐᵇ f(x) dx = F(b) − F(a); F is the antiderivative (indefinite integral).
Differentiation and integration are inverse operations: the area function's slope equals the curve's height.
Standard results: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1), ∫(1/x) dx = ln x, ∫sin x dx = −cos x, ∫cos x dx = sin x, ∫eˣ dx = eˣ.
In physics an integral adds up small contributions: distance from speed, work from force, mass from density.

Differentiation breaks a quantity into rates. Integration does the reverse: it adds up countless tiny contributions to rebuild the whole.

The area under a curve

Let a curve PQ show $y = f(x)$ between $x = a$ and $x = b$ . Drop vertical lines from P and Q to the x-axis, meeting it at A and B. What is the area of the region PABQ under the curve?

Here is a strategy that works for any curve. Chop AB into $N$ equal strips, each of width $\Delta x = (b - a)/N$ . Over each strip, draw a rectangle whose height is the curve’s height at the strip’s left edge. Rectangles are easy to measure, so add them:

I' = f(a)\Delta x + f(a + \Delta x)\Delta x + \dots + f\big(a + (N-1)\Delta x\big)\Delta x = \sum_{i=1}^{N} f(x_i)\,\Delta x

This is a little off, because thin slivers between the bar tops and the curve are missed (or overcounted). But use more, thinner strips and the bars touch the curve at more points while the slivers shrink. Try it:

Interactive

Area as a sum of thin bars

Drag the limits a and b along the x-axis. Increase N and the bars hug the curve ever more tightly; the leftover slivers shrink away.

N

\Delta x

1.1667

\sum f(x_i)\,\Delta x

17.6691

\int_a^b f\,dx

16.9430

error0.7261

Curve

Bar height taken from

Number of bars

N

As $N \to \infty$ (equivalently $\Delta x \to 0$ ) the error vanishes and the sum becomes the exact area. We write that limit with a stretched S, for “sum”:

Definite integral(2.8)

I = \int_a^b f(x)\,dx = \lim_{\Delta x \to 0}\ \sum_{i=1}^{N} f(x_i)\,\Delta x

read as “the integral of f(x) from a to b”; a is the lower limit and b the upper limit

Doing one by hand: the area under y = x

Take $f(x) = x$ . The region under it between $a$ and $b$ is a trapezium, so we know the answer should be (average height) × width $= \tfrac12(a+b)(b-a)$ . Let’s get there by summing bars. The bar heights are $a,\ a+\Delta x,\ a + 2\Delta x, \dots$ , an arithmetic progression of $N$ terms, so

I' = \frac{N}{2}\big[a + a + (N-1)\Delta x\big]\Delta x = \frac{b-a}{2}\,\big[a + b - \Delta x\big]

using $N\Delta x = b - a$ . Letting $\Delta x \to 0$ :

\int_a^b x\,dx = \frac{b-a}{2}(a+b) = \frac12\left(b^2 - a^2\right).

The shortcut: antiderivatives

Summing series by hand gets painful fast. Fortunately there is a beautiful shortcut. Find any function $F(x)$ whose derivative is $f(x)$ . Then

Fundamental theorem of calculus

\int_a^b f(x)\,dx = F(b) - F(a) \equiv \big[F(x)\big]_a^b, \quad \text{where } \frac{dF}{dx} = f(x)

$F$ is called the antiderivative or indefinite integral of $f$ , written $\int f(x)\,dx = F(x)$ . Check it on our trapezium: $\tfrac{d}{dx}(\tfrac12 x^2) = x$ , so $\int_a^b x\,dx = \tfrac12 b^2 - \tfrac12 a^2$ , exactly what the long way gave.

Interactive

Accumulating area

Drag x. The top graph shades the area under f from a to x. The bottom graph plots that area F(x) as it builds up. Notice that the slope of F at x always equals the height f(x).

x

4.00

f(x)

0.71

F(x) = \int_a^x f\,dt

6.40

\text{slope of }F

0.71

A table of integrals

f(x)	∫ f(x) dx	f(x)	∫ f(x) dx
$x^n\ (n\ne -1)$	$\dfrac{x^{n+1}}{n+1}$	$\dfrac1x$	$\ln x$
$\sin x$	$-\cos x$	$\dfrac{1}{x^2+a^2}$	$\dfrac1a\tan^{-1}\dfrac xa$
$\cos x$	$\sin x$	$\dfrac{1}{\sqrt{a^2-x^2}}$	$\sin^{-1}\dfrac xa$
$\sec^2 x$	$\tan x$	$\sec x\tan x$	$\sec x$
$\csc^2 x$	$-\cot x$	$\csc x\cot x$	$-\csc x$
$e^x$	$e^x$

And three rules:

\text{(a) } \int c f(x)\,dx = c\int f(x)\,dx \qquad \text{(b) if } \int f(x)\,dx = F(x), \text{ then } \int f(cx)\,dx = \tfrac1c F(cx)

\text{(c) } \int\big[f(x) + g(x)\big]dx = \int f(x)\,dx + \int g(x)\,dx

Worked example

A polynomial integral

Evaluate $\displaystyle\int_1^4 (3x^2 + 2x + 1)\,dx$ .

Worked example

Half a sine hump

Find the area under $y = \sin x$ from $x = 0$ to $x = \pi$ .

What to remember

1 $\int_a^b f\,dx$ is the limit of a sum of thin bars, which is the area under the curve.
2If $dF/dx = f$ , then $\int_a^b f\,dx = F(b) - F(a)$ .
3Integration and differentiation are inverse operations.
4In physics, an integral means adding up many small contributions: distance from speed, work from force, and so on.

Check yourself

∫₀² x³ dx equals…

In a left-edge Riemann sum of an increasing function, the bars…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

WatchIntegration and the fundamental theorem3Blue1Brown WatchIntegralsCrash Course Physics #3 ReadIntegration (text lesson)3Blue1Brown ReadRiemann sumWikipedia ReadIntroduction to integrationMaths is Fun ReadThe integralHyperPhysics

PreviousPeaks & Valleys Next Significant Figures

Integration: Adding Up Slivers

Slice an area into thin bars, add them all, and let the slices vanish.

16 minUpdated 21 October 2026Riemann sumdefinite integralantiderivativefundamental theorem

Lesson at a glanceShow the key points

∫ₐᵇ f(x) dx is the limit of the sum Σ f(xᵢ)Δx as Δx → 0: the area under the curve between a and b.
If dF/dx = f(x), then ∫ₐᵇ f(x) dx = F(b) − F(a); F is the antiderivative (indefinite integral).
Differentiation and integration are inverse operations: the area function's slope equals the curve's height.
Standard results: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1), ∫(1/x) dx = ln x, ∫sin x dx = −cos x, ∫cos x dx = sin x, ∫eˣ dx = eˣ.
In physics an integral adds up small contributions: distance from speed, work from force, mass from density.

Differentiation breaks a quantity into rates. Integration does the reverse: it adds up countless tiny contributions to rebuild the whole.

The area under a curve

Let a curve PQ show $y = f(x)$ between $x = a$ and $x = b$ . Drop vertical lines from P and Q to the x-axis, meeting it at A and B. What is the area of the region PABQ under the curve?

I' = f(a)\Delta x + f(a + \Delta x)\Delta x + \dots + f\big(a + (N-1)\Delta x\big)\Delta x = \sum_{i=1}^{N} f(x_i)\,\Delta x

Interactive

Area as a sum of thin bars

Drag the limits a and b along the x-axis. Increase N and the bars hug the curve ever more tightly; the leftover slivers shrink away.

N

\Delta x

1.1667

\sum f(x_i)\,\Delta x

17.6691

\int_a^b f\,dx

16.9430

error0.7261

Curve

Bar height taken from

Number of bars

N

As $N \to \infty$ (equivalently $\Delta x \to 0$ ) the error vanishes and the sum becomes the exact area. We write that limit with a stretched S, for “sum”:

Definite integral(2.8)

I = \int_a^b f(x)\,dx = \lim_{\Delta x \to 0}\ \sum_{i=1}^{N} f(x_i)\,\Delta x

read as “the integral of f(x) from a to b”; a is the lower limit and b the upper limit

Doing one by hand: the area under y = x

I' = \frac{N}{2}\big[a + a + (N-1)\Delta x\big]\Delta x = \frac{b-a}{2}\,\big[a + b - \Delta x\big]

using $N\Delta x = b - a$ . Letting $\Delta x \to 0$ :

\int_a^b x\,dx = \frac{b-a}{2}(a+b) = \frac12\left(b^2 - a^2\right).

The shortcut: antiderivatives

Summing series by hand gets painful fast. Fortunately there is a beautiful shortcut. Find any function $F(x)$ whose derivative is $f(x)$ . Then

Fundamental theorem of calculus

\int_a^b f(x)\,dx = F(b) - F(a) \equiv \big[F(x)\big]_a^b, \quad \text{where } \frac{dF}{dx} = f(x)

Interactive

Accumulating area

Drag x. The top graph shades the area under f from a to x. The bottom graph plots that area F(x) as it builds up. Notice that the slope of F at x always equals the height f(x).

x

4.00

f(x)

0.71

F(x) = \int_a^x f\,dt

6.40

\text{slope of }F

0.71

A table of integrals

f(x)	∫ f(x) dx	f(x)	∫ f(x) dx
$x^n\ (n\ne -1)$	$\dfrac{x^{n+1}}{n+1}$	$\dfrac1x$	$\ln x$
$\sin x$	$-\cos x$	$\dfrac{1}{x^2+a^2}$	$\dfrac1a\tan^{-1}\dfrac xa$
$\cos x$	$\sin x$	$\dfrac{1}{\sqrt{a^2-x^2}}$	$\sin^{-1}\dfrac xa$
$\sec^2 x$	$\tan x$	$\sec x\tan x$	$\sec x$
$\csc^2 x$	$-\cot x$	$\csc x\cot x$	$-\csc x$
$e^x$	$e^x$

And three rules:

\text{(a) } \int c f(x)\,dx = c\int f(x)\,dx \qquad \text{(b) if } \int f(x)\,dx = F(x), \text{ then } \int f(cx)\,dx = \tfrac1c F(cx)

\text{(c) } \int\big[f(x) + g(x)\big]dx = \int f(x)\,dx + \int g(x)\,dx

Worked example

A polynomial integral

Evaluate $\displaystyle\int_1^4 (3x^2 + 2x + 1)\,dx$ .

Worked example

Half a sine hump

Find the area under $y = \sin x$ from $x = 0$ to $x = \pi$ .

What to remember

1 $\int_a^b f\,dx$ is the limit of a sum of thin bars, which is the area under the curve.
2If $dF/dx = f$ , then $\int_a^b f\,dx = F(b) - F(a)$ .
3Integration and differentiation are inverse operations.
4In physics, an integral means adding up many small contributions: distance from speed, work from force, and so on.

Check yourself

∫₀² x³ dx equals…

In a left-edge Riemann sum of an increasing function, the bars…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

PreviousPeaks & Valleys Next Significant Figures