Level 5 · Mathematics M5.5

Integration

Undo differentiation to find areas under curves and totals from rates.

Planned. This book is not written yet. Below is its plan, which a teacher can already use.

You are ready for this book if

You can differentiate 3x² − 4x + 1.

The plan

  1. M5.5.1 Undoing differentiation
    • Integrate xⁿ for any n except −1
    • Explain why every answer needs a constant + c
  2. M5.5.2 Definite integrals and area
    • Find the area under a curve with a definite integral
    • Find an area that lies below the x axis
  3. M5.5.3 Area between curves
    • Find where a line and a curve meet to set the limits
    • Find the area between a line and a curve
  4. M5.5.4 More functions to integrate
    • Integrate eˣ, 1/x, sin x and cos x
    • Integrate e³ˣ and cos 2x
  5. M5.5.5 Substitution and integration by parts
    • Integrate x(x² + 1)⁴ by substitution
    • Integrate x eˣ by parts
    • Choose between substitution and parts for an integral
  6. M5.5.6 Estimating areas
    • Estimate an integral with the trapezium rule
    • Say if the estimate is too big or too small, and why
  7. M5.5.7 Differential equations
    • Solve dy/dx = ky by separating the variables
    • Model cooling or growth with a differential equation
  8. M5.5.8 Motion and totals
    • Find distance from a velocity that changes with time
    • Find the total water that flows from a rate of flow

For parents and teachers

Level 5 fits learners who can already: read a textbook. use functions and graphs. In many schools that is about 17 to 20, but start wherever the learner is.

Strand: Algebra. Written and drawn by attnom. Free to read, print and share.