Chapter 8 Integral Calculus

Section 8.3 Applications

8.3.3 Applications in the Sciences

The velocity $v$ of an object describes the instantaneous rate of change of position at the time $t$. Thus, we have $v=\frac{ds}{dt}$ if $v=v(t)$ and $s=s(t)$ are considered as functions of $t$. The current position $s(T)$ results from the inversion of the derivative, i.e. by integration of the velocity over the time $t$. With the initial value $s(t=0)=s_0$ at the time $t=0$, we have

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{{\int }_0^T\frac{ds}{dt}\hspace{0.17em}dt}& =\hfill & {\int }_0^{T}v\hspace{0.17em}dt\hfill \\ \multicolumn{1}{c}{\iff \mathrm{\hspace{0.5em}\hspace{0.5em}}\mathrm{\hspace{0.5em}\hspace{0.5em}}{[s(t)]}_0^T}& =\hfill & {\int }_0^{T}v\hspace{0.17em}dt\hfill \\ \multicolumn{1}{c}{\iff \mathrm{\hspace{0.5em}\hspace{0.5em}}\mathrm{\hspace{0.5em}\hspace{0.5em}}s(T)-s(0)}& =\hfill & {\int }_0^{T}v\hspace{0.17em}dt\hfill \\ \multicolumn{1}{c}{\iff \mathrm{\hspace{0.5em}\hspace{0.5em}}\mathrm{\hspace{0.5em}\hspace{0.5em}}s(T)}& =\hfill & s_0+{\int }_0^{T}v(t)\hspace{0.17em}dt\hspace{0.5em}.\hfill \end{array}}$
In mathematical terms, this situation can be described as follows: if the derivative $f\text{'}$ of a function $f$ and a single function value $f(x_0)$ are known, then the function can by calculated by means of the integral. In this context one says that the function values are reconstructed from the derivative.
If, for example, a population of bacteria increases approximately according to $B\text{'}$ with $B\text{'}(t)=0.6t$ for $t\ge 0$ and initially the population consists of $B(0)=100$ bacteria, then the number $B$ of bacteria in the population at time $T$ is described by

${\displaystyle B(T)-B(0)={\int }_0^{T}0.6t\hspace{0.17em}dt}$
and hence by

${\displaystyle B(T)=B(0)+{\int }_0^{T}0.6t\hspace{0.17em}dt=100+0.6{\int }_0^{T}t\hspace{0.17em}dt=100+0.3(T^2-0^2)=100+0.3T^2\hspace{0.5em}.}$
Thus, the fundamental theorem of calculus provides an important tool for reconstructing a function if its derivative is known (and continuous). However, in practical applications the functions will often be more sophisticated, for example consisting of combinations of exponential functions.
A further example from physics, which may be familiar, is the determination of the work as a product of force and displacement: $W=F_s·s$. Here, $F_s$ is the projection of the force onto the direction of the travelled path. However, if the force depends on the path, then this law does not apply in its simple form. For example, to calculate the work done by moving a massive body along a path, the force has to be integrated along the path from the initial point $s_1$ to the end point $s_2$:

${\displaystyle \begin{array}{c}\multicolumn{1}{c}{W={\int }_{s_1}^{s_2}{F}_s(s)\hspace{0.17em}ds\hspace{0.5em}.}\end{array}}$
These are only three examples from the sciences of how the notion of an integral is useful. Depending on the subject of your study you will encounter a whole series of further applications of integration.