INTEGRATION 09

Applications of Integration

Integration turns rates and local contributions into total quantities.

The main idea

If a quantity is built continuously from small contributions, integration can often recover the total.

\[ \boxed{ \text{local contribution} \longrightarrow \text{integral} \longrightarrow \text{total quantity} } \]

1. Area under a curve

\[ f(x)\ge0 \]

on \([a,b]\), then

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

gives the area between the graph of \(y=f(x)\), the \(x\)-axis, and the vertical lines \(x=a\) and \(x=b\).

2. Area between two curves

Suppose

\[ f(x)\ge g(x) \]

on \([a,b]\).

Then the area between the curves is

\[ \boxed{ A= \int_a^b \left[f(x)-g(x)\right]\,dx } \]

The vertical distance between the curves is

\[ f(x)-g(x). \]

Integration accumulates these small vertical strips across the interval.

3. Example: area between two curves

Consider

\[ f(x)=x \]

and

\[ g(x)=x^2 \]

between \(x=0\) and \(x=1\).

Since \(x\ge x^2\) on this interval,

\[ A= \int_0^1(x-x^2)\,dx. \]

Therefore,

\[ A = \left[ \frac{x^2}{2} - \frac{x^3}{3} \right]_0^1. \]

\[ A = \frac12-\frac13 = \frac16. \]

Hence,

\[ \boxed{ A=\frac16 } \]

4. Displacement from velocity

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

then displacement from time \(a\) to time \(b\) is

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

Integration recovers total position change from the rate of change of position.

5. Example: constant velocity

\[ v(t)=4 \]

for \(0\le t\le3\), then

\[ \int_0^3 4\,dt = 12. \]

So the displacement is \(12\) units.

6. Variable velocity

Suppose

\[ v(t)=2t. \]

From \(t=0\) to \(t=3\),

\[ s(3)-s(0) = \int_0^3 2t\,dt. \]

Therefore,

\[ = [t^2]_0^3 = 9. \]

So the total displacement is

\[ \boxed{9} \]

units.

7. Distance versus displacement

A velocity can be negative.

Then

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

gives signed displacement.

To obtain total distance travelled, integrate the speed:

\[ \boxed{ \text{distance} = \int_a^b |v(t)|\,dt } \]

8. Accumulating a rate

The same principle works for any rate.

\[ r(t)=\frac{dQ}{dt}, \]

then

\[ Q(b)-Q(a) = \int_a^b r(t)\,dt. \]

This can represent population change, mass accumulation, fluid flow, energy transfer or many other quantities.

9. Example: population growth rate

Suppose a population changes at rate

\[ \frac{dP}{dt}=100e^{0.2t}. \]

The population change between times \(0\) and \(T\) is

\[ P(T)-P(0) = \int_0^T100e^{0.2t}\,dt. \]

Since

\[ \int e^{0.2t}\,dt = 5e^{0.2t}, \]

we obtain

\[ P(T)-P(0) = 500[e^{0.2t}]_0^T. \]

Therefore,

\[ \boxed{ P(T)-P(0) = 500(e^{0.2T}-1) } \]

10. Average value of a function

The average value of \(f(x)\) over \([a,b]\) is

\[ \boxed{ f_{\text{avg}} = \frac{1}{b-a} \int_a^b f(x)\,dx } \]

The integral gives the total accumulation, and dividing by the interval length gives the average per unit of \(x\).

11. Example: average value

Find the average value of

\[ f(x)=x^2 \]

\[ [0,2]. \]

Then

\[ f_{\text{avg}} = \frac{1}{2} \int_0^2 x^2\,dx. \]

\[ = \frac12 \left[ \frac{x^3}{3} \right]_0^2 = \frac12\cdot\frac83. \]

Therefore,

\[ \boxed{ f_{\text{avg}}=\frac43 } \]

12. Volume by cross-sectional area

Suppose a solid has cross-sectional area \(A(x)\) at position \(x\).

A thin slice of thickness \(dx\) has approximate volume

\[ dV=A(x)\,dx. \]

Accumulating the slices gives

\[ \boxed{ V= \int_a^b A(x)\,dx } \]

13. Volume of revolution

If the graph \(y=f(x)\) is rotated around the \(x\)-axis, each cross-section is a disk of radius \(f(x)\).

Its area is

\[ \pi[f(x)]^2. \]

Therefore,

\[ \boxed{ V= \pi\int_a^b [f(x)]^2\,dx } \]

14. Probability density

For a continuous random variable with probability density \(p(x)\),

\[ p(x)\ge0 \]

and

\[ \int_{-\infty}^{\infty}p(x)\,dx=1. \]

The probability that \(X\) lies between \(a\) and \(b\) is

\[ \boxed{ P(a\le X\le b) = \int_a^b p(x)\,dx } \]

So probability can also be interpreted as accumulated density.

15. The common structure

These applications may look different, but mathematically they share the same idea:

\[ \boxed{ \text{total} = \int \text{rate or density} \times \text{small interval} } \]

Examples include

\[ \text{displacement} = \int \text{velocity}\,dt, \] \[ \text{population change} = \int \text{growth rate}\,dt, \] \[ \text{volume} = \int \text{cross-sectional area}\,dx, \] \[ \text{probability} = \int \text{probability density}\,dx. \]

16. Differentiation and integration together

Differentiation moves from a total quantity to its local rate:

\[ Q(t) \longrightarrow \frac{dQ}{dt}. \]

Integration moves from the rate back to accumulated change:

\[ \frac{dQ}{dt} \longrightarrow \int\frac{dQ}{dt}\,dt. \]

\[ \boxed{ \text{quantity} \overset{\text{differentiate}}{\longrightarrow} \text{rate} \overset{\text{integrate}}{\longrightarrow} \text{change in quantity} } \]

17. Why this matters for modelling

Mathematical models often describe how quickly something changes rather than giving the total quantity directly.

Integration converts that rate information into accumulated effects.

This idea will become central when we study differential equations.