- Understand and apply the net change theorem to calculate how quantities change over an interval
- Use integration formulas to calculate the integrals of odd and even functions
The Net Change Theorem
The Main Idea
- Theorem Statement:
- The new value of a changing quantity equals the initial value plus the integral of the rate of change
- [latex]F(b) = F(a) + \int_a^b F'(x) dx[/latex] Alternatively: [latex]\int_a^b F'(x) dx = F(b) - F(a)[/latex]
- Key Implications:
- Relates the integral of a rate of change to the total change in a quantity
- Applies to various physical quantities: displacement, fluid flow, population growth, etc.
- Automatically accounts for both positive and negative changes
- Applications:
- Displacement from velocity
- Total distance traveled
- Fluid consumption or accumulation
- Net change in any quantity with a known rate of change
- Net Change vs. Total Change:
- Net change can be positive, negative, or zero
- Total change is always positive (use absolute value of rate)
Find the net displacement and total distance traveled in meters given the velocity function [latex]f(t)=\frac{1}{2}{e}^{t}-2[/latex] over the interval [latex]\left[0,2\right].[/latex]
A particle moves along a straight line with velocity function [latex]v(t) = t^2 - 4t + 3[/latex] m/s, where t is in seconds. Find:
- The net displacement of the particle from [latex]t = 0[/latex] to [latex]t = 5[/latex] seconds.
- The total distance traveled by the particle during this time interval.
Integrating Even and Odd Functions
The Main Idea
- Even Functions:
- Definition: [latex]f(-x) = f(x)[/latex] for all [latex]x[/latex] in the domain
- Symmetric about the [latex]y[/latex]-axis
- For continuous even functions: [latex]\int_{-a}^a f(x) dx = 2\int_0^a f(x) dx[/latex]
- Odd Functions:
- Definition: [latex]f(-x) = -f(x)[/latex] for all [latex]x[/latex] in the domain
- Symmetric about the origin
- For continuous odd functions: [latex]\int_{-a}^a f(x) dx = 0[/latex]
- Geometric Interpretation:
- Even functions: Equal areas on both sides of [latex]y[/latex]-axis
- Odd functions: Equal and opposite areas on either side of [latex]y[/latex]-axis
- Applications:
- Simplifying calculations for definite integrals
- Proving certain integrals are zero without calculation
- Useful in Fourier analysis and other advanced mathematical topics
Integrate the function [latex]{\displaystyle\int }_{-2}^{2}{x}^{4}dx.[/latex]
Evaluate [latex]\int_{-\pi}^{\pi} (x^4 + \cos x) dx[/latex] using properties of even and odd functions.