Applications of Integration: Average Value and Particle Motion

Average Value of a Function

The distinct height a function would take if it were a flat, constant horizontal line while maintaining the same area under the curve.

Average Value Formula

favg = (1/(b-a)) * ∫a^b f(x) dx, where f is integrable on [a,b].

Area under the curve

∫_a^b f(x) dx, represents the total area between the function and the x-axis over the interval [a,b].

Mean Value Theorem for Integrals (MVT-I)

States that if f is continuous on [a,b], then there exists at least one number c in (a,b) such that f(c) = f_avg.

Displacement

The net change in position, represented as ∫_t1^t2 v(t) dt.

Total Distance Traveled

The total distance accumulated regardless of direction, calculated using ∫_t1^t2 |v(t)| dt.

Net Change Theorem

A principle used to relate position, velocity, and acceleration using integrals.

s(t)

Represents position as a function of time t.

v(t)

Represents velocity as a function of time t.

a(t)

Represents acceleration as a function of time t.

Final Position Formula

s(b) = s(a) + ∫_a^b v(t) dt.

Average Rate of Change

Calculated as (f(b)-f(a))/(b-a).

Particle Turnaround Point

The point where the velocity function v(t) equals zero, indicating a change in direction.

Height in Average Value

Expressed as Height = Area/Width, where Area = ∫ and Width = (b-a).

Continuous Function

A function that does not have any abrupt changes or discontinuities in its domain.

Velocity Function

v(t) = s'(t), the derivative of the position function.

Acceleration Function

a(t) = v'(t) = s''(t), the derivative of the velocity function.

Speed

The absolute value of velocity, |v(t)|.

Area vs Width

In the average value context, Area is integrated area under the curve and Width is the interval length (b-a).

Fundamental Theorem of Calculus

Establishes the relationship between differentiation and integration, providing a way to evaluate integrals.

Example of Average Value

To find the average value of f(x) = 3x^2 - 2x on [1, 4], integrate and use the average value formula.

Total Distance Calculation

Requires checking where the velocity function changes sign and using absolute values in the integral.

Initial Position in Motion Problems

Always included in final position calculations as s(b) = s(0) + ∫v(t) dt.

Critical Distinction

The difference between displacement (vector) and total distance (scalar) in motion problems.

Integrable Function

A function for which the integral can be computed over a specific interval.

Flipping Between Derivatives and Integrals

Moving 'backwards' from acceleration to velocity or from velocity to position relies on integration.

Average Value of Velocity

Is the Average Value of the velocity function, related to position change.