Integration 2

Summary

Integration is the reverse process of differentiation and is used to find areas under curves and between curves. It involves evaluating definite integrals and using them to calculate areas and volumes of revolution.

• Definite Integrals — used to calculate the area under a curve between two points. Example: $\int_{a}^{b} f(x) \, dx$ gives the area under $f(x)$ from $x = a$ to $x = b$.
• Areas Under a Curve — calculated by summing the areas of rectangles under the curve. Example: The area under $f(x)$ from $x = a$ to $x = b$ is $\int_{a}^{b} f(x) \, dx$.
• Areas Under the x-axis — similar to areas under a curve but result in negative values. Example: The area from $x = 0$ to $x = 3$ under the x-axis is negative.
• Areas Between Curves and Lines — found by subtracting the area under one curve from another. Example: $\int_{a}^{b} [f(x) - g(x)] \, dx$ gives the area between $f(x)$ and $g(x)$.
• The Trapezium Rule — approximates the area under a curve by dividing it into trapezoids. Example: Used to estimate $\int_{a}^{b} f(x) \, dx$ by calculating the area of trapezoids.

Exam Tips

Key Definitions to Remember

• Definite integrals calculate the area under a curve between two points.
• The trapezium rule approximates the area under a curve using trapezoids.

Common Confusions

• Forgetting to subtract areas when finding the area between two curves.
• Mixing up positive and negative areas under the x-axis.

Typical Exam Questions

• How do you find the area under a curve from $x = a$ to $x = b$? Use $\int_{a}^{b} f(x) \, dx$.
• How do you calculate the area between two curves? Subtract the integral of the lower curve from the upper curve.
• How is the trapezium rule applied? Divide the interval into trapezoids and sum their areas.

What Examiners Usually Test

• Ability to evaluate definite integrals.
• Understanding of how to find areas between curves and lines.
• Application of the trapezium rule for area approximation.