The indefinite integral applications, Kinetic equation of free fall, Exponential growth and decay (increase and decrease)

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Integral calculus

The indefinite integral applications

Kinetic equation of free fall

In physics, physical laws are described by mathematical formulas. For example,

kinetic quantity velocity v, defined as the rate at which position s changes with respect to time t,

represents an antiderivative or primitive function of v(t), hence

Since, at t = t₀, the formula for displacement gives s(t₀) = 0 then, the constant t₀ denotes the point in time when starts measurement of the displacement. Thus for example,

the velocity of an object in free fall motion

v = gt,

where g = 9,81 m/s/s is the acceleration of free fall (the acceleration due to gravity),

therefore the displacement

what can be checked over by differentiating the right-hand side with respect to t to obtain the integrand gt.

Exponential growth and decay (increase and decrease)

A quantity that increases or decreases with respect to time t, or other argument, such that the rate of change by which it happens at an instant t is proportional by the value of the quantity at that instant, then it changes by the exponential law.

Let x(t) denotes the value of a quantity at a time t, the rate dx/dt, at which the quantity changes at an instant t is proportional by the value of the quantity at that instant, with constant of proportionality k then, x(t) satisfies the differential equation of the exponential function

where the sign of constant of proportionality k depends on whether x(t) increases or decreases with t.

To solve the above differential equation we rewrite it by multiplying by dt and dividing by x(t) to separate variables so that each side of the equation can be integrated with respect to each variable,

Setting t = 0 to both sides of the solution gives | x(0) | = C thus, the constant of integration defines the initial value of the quantity |x(t)|.

Therefore, the general solution of the differential equation

dx/dt = k · x(t)

is represented by the exponential function multiplied by an arbitrary constant x(0),

x(t) = x(0) e^kt

or y(t) = C e^kt is a solution to the differential equation dy/dt = k y.

Recall that the graph of the exponential function f(x) = e^xand its derivative coincide, i.e.,

y' = dy/dx = e^x = y

so that the sub tangent (projection of the segment of the tangent onto the x-axis) drawn at any point x,

s_t = y / y' = 1, as is shown in the above figure.

Therefore, for x(0) > 0 and k > 0, the quantity x(t) increases as t increases while, in case the coefficient of t is negative, i.e., k < 0, x(t) decreases, for which an example is radioactive decay. In this case the graph of the exponential function reflects around the y-axis.