- 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
- F(b)=F(a)+∫baF′(x)dx Alternatively: ∫baF′(x)dx=F(b)−F(a)
- 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 f(t)=12et−2 over the interval [0,2].
A particle moves along a straight line with velocity function v(t)=t2−4t+3 m/s, where t is in seconds. Find:
- The net displacement of the particle from t=0 to t=5 seconds.
- The total distance traveled by the particle during this time interval.
Integrating Even and Odd Functions
The Main Idea
- Even Functions:
- Definition: f(−x)=f(x) for all x in the domain
- Symmetric about the y-axis
- For continuous even functions: ∫a−af(x)dx=2∫a0f(x)dx
- Odd Functions:
- Definition: f(−x)=−f(x) for all x in the domain
- Symmetric about the origin
- For continuous odd functions: ∫a−af(x)dx=0
- Geometric Interpretation:
- Even functions: Equal areas on both sides of y-axis
- Odd functions: Equal and opposite areas on either side of y-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 ∫2−2x4dx.
Evaluate ∫π−π(x4+cosx)dx using properties of even and odd functions.