INTEGRATION 01

Integration as Accumulation

Integration measures how small contributions build up to produce a total quantity.

The main idea

Many quantities are formed by accumulating small pieces.

Integration provides a mathematical way to combine these contributions.

\[ \boxed{ \text{small contributions} \longrightarrow \text{total accumulation} } \]

1. A simple discrete example

Suppose four contributions are

\[ 3,\qquad 5,\qquad 2,\qquad 4. \]

The total is

\[ 3+5+2+4=14. \]

This is ordinary finite addition.

2. When contributions vary continuously

Now imagine that contributions occur continuously rather than at only a few separate points.

For example, a velocity may change continuously with time.

During a short time interval \(\Delta t\), the distance travelled is approximately

\[ \text{velocity}\times\text{time interval}. \]

\[ \Delta s\approx v(t)\Delta t. \]

3. Add the small distances

Divide a time interval into many small pieces.

The total distance is approximately

\[ v(t_1)\Delta t + v(t_2)\Delta t + v(t_3)\Delta t +\cdots. \]

Using summation notation,

\[ \sum_i v(t_i)\Delta t. \]

4. Make the intervals smaller

The approximation becomes more accurate as the time intervals become smaller.

Integration studies the limiting value as

\[ \Delta t\to0. \]

This leads to

\[ \boxed{ \int_a^b v(t)\,dt } \]

which gives the accumulated effect of the rate \(v(t)\) from \(t=a\) to \(t=b\).

5. What the integral symbol means

The symbol

\[ \int \]

is an elongated form of the letter \(S\), historically connected with the idea of a sum.

So an integral may be understood as a continuous form of summation.

6. Reading the notation

The expression

\[ \int_a^b f(x)\,dx \]

is read as

“the integral of \(f(x)\) with respect to \(x\), from \(a\) to \(b\)”.

Here:

\[ a \]

is the lower limit,

\[ b \]

is the upper limit,

\[ f(x) \]

is the quantity being accumulated,

and

\[ dx \]

indicates the variable of integration.

7. Area interpretation

\[ f(x)\ge0, \]

then

\[ \int_a^b f(x)\,dx \]

can be interpreted as the area under the graph of \(y=f(x)\) between \(x=a\) and \(x=b\).

8. Integration is broader than area

Area is one important interpretation, but integration is more general.

If \(v(t)\) is velocity, then

\[ \int_a^b v(t)\,dt \]

gives accumulated displacement.

If \(r(t)\) is a rate of population growth, then

\[ \int_a^b r(t)\,dt \]

gives the total population change over that interval.

9. Rate and accumulated quantity

Differentiation and integration connect rates with totals.

\[ \boxed{ \text{rate} \quad\longleftrightarrow\quad \text{accumulated quantity} } \]

For example,

\[ v(t)=\frac{ds}{dt} \]

describes the rate at which position changes.

Integrating velocity accumulates those changes:

\[ s(b)-s(a) = \int_a^b v(t)\,dt. \]

10. Why small intervals matter

For a varying function, a rectangle of width \(\Delta x\) and height \(f(x_i)\) has approximate area

\[ f(x_i)\Delta x. \]

Adding many rectangles gives

\[ \sum_i f(x_i)\Delta x. \]

As the largest interval width approaches zero,

\[ \Delta x\to0, \]

the approximation approaches the exact integral.

11. The conceptual transition

\[ \text{finite sum} \] \[ \downarrow \] \[ \sum_i f(x_i)\Delta x \] \[ \downarrow\quad \Delta x\to0 \] \[ \boxed{ \int_a^b f(x)\,dx } \]

Integration is therefore built from the same limiting idea that underlies differentiation.

12. Where this leads

The next question is how we calculate an integral without constructing infinitely many rectangles every time.

The answer comes from the connection between integration and differentiation.

\[ \boxed{ \text{differentiation} \longleftrightarrow \text{integration} } \]

This leads to antiderivatives and the Fundamental Theorem of Calculus.