Comprehensive Guide to Applications of Integration (Unit 8)

Average Rate of Change

The slope of the secant line in differential calculus.

Average Value

The height of a rectangle whose base is (b-a) and whose area equals the area under the curve f(x).

Average Value Formula

f_{avg} = (1/(b-a)) * ∫[a to b] f(x) dx.

Mean Value Theorem for Integrals

States there exists a number c in [a,b] such that f(c) = f_{avg}.

Displacement

Net change in position, calculated as ∫[a to b] v(t) dt.

Total Distance Traveled

The sum of all absolute movements, calculated as ∫[a to b] |v(t)| dt.

Vertical Slicing

Finding area between curves with respect to x using A = ∫[a to b] [f(x) - g(x)] dx.

Horizontal Slicing

Finding area between curves with respect to y using A = ∫[c to d] [f(y) - g(y)] dy.

Volume of a Solid

V = ∫[a to b] A(x) dx, where A(x) is the area of cross sections.

Disc Method

Used for volumes when the region is flush against the axis of rotation: V = π ∫[a to b] [R(x)]² dx.

Washer Method

Used for volumes when there is a gap: V = π ∫a to b]² - [r(x)]²) dx.

Arc Length Formula (y=f(x))

L = ∫[a to b] √(1 + [f'(x)]²) dx.

Common Mistake: Displacement vs. Total Distance

Check if the problem asks for the net change or total distance traveled.

Common Mistake: Washer Method

Remember to use (R² - r²), not (R - r)².

Position Function

s(t) represents position in motion along a line.

Velocity Function

v(t) = ∫ a(t) dt + C, where C can be found from an initial condition.

Position from Velocity Formula

s(b) = s(a) + ∫[a to b] v(t) dt.

Area between curves

A = ∫[a to b] [f(x) - g(x)] dx when f(x) ≥ g(x).

Cautions with Cross Sections

Identify the area formula for the type of cross section being used.

Semi-Circle Area Formula

A = (π/8) s² for semicircular cross sections.

Equilateral Triangle Area Formula

A = (√3/4) s² for equilateral triangular cross sections.

Finding Intersection Points

Set f(x) = g(x) to determine limits of integration.

Rotation around y=k

Adjust radius to (f(x) - k) or (k - f(x)).

Integrable Function

A function f that can be integrated over a closed interval [a, b].

Velocity Interpretation

Indicates speed and direction of motion at any given time.

Acceleration Function

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

Average Value Geometric Interpretation

The average value of f(x) gives the height of a rectangle under the curve.

Continous Functions in MVT

If f is continuous on [a, b], MVT applies.

Example of Displacement Calculation

Displacement for v(t) = t² - 4 over [0, 3] is ∫[0 to 3] (t² - 4) dt.

Example of Total Distance Calculation

Total distance involves splitting the integral when v(t)=0.

Integral of Absolute Value

Used to find total distance traveled from velocity.

3D Solids and Cross Sections

A 3D solid formed by rotation of a 2D area; area of the base integrated for volume.

Using π in Volumes of Revolution

Always include π in the volume integral for disks/washers.

Critical for Integration Variables

Ensure that variable limits match the integration variable.

Shell Method

An alternative method for finding volumes when revolving about an axis.

Hypotenuse Formula in Arc Length

dl = √(dx² + dy²) for calculating arc lengths.

Fundamental Relationships in Motion

Velocity and position are related through integration.

Mean Value Property of Integrals

Indicates that the function achieves its average value at least once.

Key Functions in Motion

Position, Velocity, Acceleration are interconnected functions.

Integral Boundaries

Boundaries of integration are determined by the nature of the problem.

Dimensions of Area

Differentiating between dimensions depending on the slicing method used.

Application of Accumulation Principle

Used for calculating the area between curves using integration.