Overview
This lecture covers the Net Change Theorem, demonstrates its application in various problems, and distinguishes between displacement and total distance in integral calculus.
Net Change Theorem
- The Net Change Theorem states: ∫ₐᵇ F'(t) dt = F(b) − F(a), where F'(t) is the derivative of F(t).
- This represents the net change in a quantity F from time a to time b.
- The definite integral of a rate of change over an interval gives the total accumulation over that interval.
Applications & Examples
Leaky Tank Example
- Given a water leak rate r(t) = 18/(1 + t²), the total water leaked from t = 0 to t = 10 is ∫₀¹⁰ r(t) dt.
- The integral evaluates to 18 [arctan(10) - arctan(0)] ≈ 26.48 gallons.
Displacement vs. Distance
- Displacement: ∫ₐᵇ v(t) dt, where v(t) is velocity, gives net change in position (can be negative).
- Distance: ∫ₐᵇ |v(t)| dt gives total path length traversed (always positive).
- Example with v(t) = t² − 3t from t = 2 to t = 4:
- Displacement = 2/3 meters.
- Distance traveled = 3 meters (requires splitting integral and using absolute values).
Oscillating Mass Example
- For v(t) = 2sin(t) + 2cos(t), from t = 0 to t = 2π:
- Displacement: ∫₀²π v(t) dt = 0 (returns to start).
- Distance: Add positive values over intervals split at zeros of v(t); total is 8√2.
Work Done by a Force
- Work, W, done by force F(x) from x = a to x = b: W = ∫ₐᵇ F(x) dx.
- Example: F(x) = x² + x − 2, from x = 2 to x = 8, compute W = ∫₂⁸ (x²+x−2) dx ≈ 168.375 units.
Reformulation of the Net Change Theorem
- The quantity at time B: F(b) = F(a) + ∫ₐᵇ F'(t) dt.
- This represents accumulation: start at F(a), add up all instantaneous changes.
Example with Abstract Function
- Given F(1) = 3, F'(t) = 1/√t, find F(16).
- Compute F(16) = 3 + ∫₁¹⁶ 1/√t dt = 3 + 6 = 9.
Key Terms & Definitions
- Net Change Theorem — The definite integral of a derivative over [a, b] equals F(b) - F(a).
- Displacement — Net change in position; ∫ₐᵇ v(t) dt.
- Distance — Total path length; ∫ₐᵇ |v(t)| dt.
- Work — Total energy transferred; ∫ₐᵇ F(x) dx.
- Antiderivative — Function whose derivative is the given function.
Action Items / Next Steps
- Practice solving definite integrals with and without absolute values.
- Review homework problems involving accumulation and net change.
- Read textbook sections on Fundamental Theorem of Calculus and applications.