Overview
This lecture covers the AS pure mathematics chapter on integration, including finding indefinite and definite integrals, areas under curves, handling areas under the x-axis, and areas between curves and lines.
Introduction to Integration
- Integration is the reverse process of differentiation.
- To integrate, increase the power of x by one, then divide the coefficient by the new power.
- The integral sign ∫ is used, with dx specifying the variable of integration.
- After integrating, always add "+ C" (constant of integration) for indefinite integrals.
Indefinite Integrals and the Constant of Integration
- Indefinite integrals represent a family of functions differing by a constant.
- The constant "+ C" accounts for any constant term that disappears when differentiating.
- If given a specific point (x, y), substitute to solve for C.
Practice: Integrating Polynomials and Fractions
- Increase the power of each term by one and divide by the new power.
- For fractional exponents, add 1 (convert to common denominator if needed) before dividing.
- Always simplify results and remember "+ C".
Solving with Initial Conditions
- Expand complex numerators and divide each term by the denominator to simplify before integrating.
- Integrate each term, add "+ C", and substitute given values to find C.
- Write the final answer with C substituted back in.
Definite Integrals and Area
- Definite integrals find the net area between the curve and the x-axis over an interval [a, b].
- Integrate the function, substituting both upper and lower limits, then subtract.
- For definite integrals, "+ C" cancels out and is omitted.
Area Under Curves Using Definite Integrals
- To find the area bounded by a curve and the x-axis, integrate between the given limits.
- If limits include zero, substituting zero often simplifies calculations.
Area Under the X-Axis
- When integrating over intervals below the x-axis, results are negative.
- Convert negative areas to positive values when reporting total area.
- For areas above and below the axis, integrate each section separately and sum their absolute values.
Area Between a Curve and a Line
- Find intersection points by equating the curve and the line, solve for x.
- Area between a curve and a line = area under the line over the interval minus area under the curve.
- Use definite integrals for both the curve and the line over the same interval and subtract.
Key Terms & Definitions
- Integration — The reverse operation of differentiation; finds the original function.
- Indefinite Integral — The general solution including "+ C"; represents a family of functions.
- Definite Integral — Computes the net area under a curve between specified limits; gives a numerical value.
- Constant of Integration (+ C) — An unknown constant added to indefinite integrals.
- Finite Region — The bounded area between curves or between a curve and a line.
Action Items / Next Steps
- Practice integrating polynomials and fractional exponents.
- Solve problems involving initial conditions to find C.
- Find areas using definite integrals, including cases under and above the x-axis.
- Look up and review the next chapter in the curriculum for further study.