Overview
This lecture covers the fundamentals of integration in AS Pure Mathematics, including finding indefinite and definite integrals, using integration to determine areas under and between curves, and dealing with areas below the x-axis.
Introduction to Integration
- Integration is the reverse process of differentiation.
- To integrate, increase the power of x by one and divide by the new power.
- Always add a constant of integration (+C) to indefinite integrals because the original function could have any constant value.
- The integral sign ∫ is used to denote integration with respect to x (dx).
Indefinite Integrals and the Constant of Integration
- Integrating 2x gives x² + C because differentiating any x² plus a constant yields 2x.
- The constant C represents an unknown constant that disappears during differentiation.
- If specific x and y values are given, substitute them to solve for C.
Integrating Complex Expressions
- Expand brackets and simplify expressions before integrating if needed.
- Integrate each term separately, apply the power rule, and simplify results.
- When integrating terms with fractional powers, add one to the power, then divide by the new power.
Using Initial Conditions to Find C
- Substitute given points (x, y values) into the integrated function to solve for the constant C.
- Rewrite the final answer by plugging the value of C back into the integrated expression.
Definite Integrals
- Definite integrals use upper and lower limits to find the net change or area under a curve between two points.
- No +C is needed because it cancels out during subtraction.
- Apply the limits by substituting them into the integrated function and subtracting the lower from the upper result.
Area Under and Between Curves
- The definite integral from a to b of a function gives the area between the curve and the x-axis.
- If the curve goes beneath the x-axis over an interval, the definite integral yields a negative result; use the absolute value for the area.
- To find the area between a curve and a line, calculate the area under each and subtract appropriately.
Key Terms & Definitions
- Integration — The reverse process of differentiation; finds the original function from its derivative.
- Indefinite Integral — An integral without limits; includes a constant of integration (C).
- Definite Integral — An integral with upper and lower limits; calculates the net area under a curve between these points.
- Constant of Integration (C) — The unknown constant added to indefinite integrals.
- Area under a curve — The region bounded by the curve, the x-axis, and the interval [a, b].
Action Items / Next Steps
- Practice integrating polynomials and functions with fractional powers.
- Complete assigned exercises on definite and indefinite integrals.
- Review how to find the area between curves and the x-axis using definite integrals.