Integral calculus: accumulation, methods and applications

Lesson sequence

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1. Build accumulation from small contributions

   Lesson details and objectives

   Use partitions, sample points and units to distinguish a finite estimate from a limiting integral.

   • Construct left, right and midpoint sums.
   • Interpret a sum of rate times interval width.
   • Explain the mesh-size condition and finite-data limitations.

   6 practice tasks.

2. Signed area and integral bookkeeping

   Lesson details and objectives

   Apply orientation, splitting, linearity and comparison without mistaking net change for total area.

   • Use definite-integral properties.
   • Distinguish integral from geometric area.
   • Apply comparison only with pointwise ordering and correct orientation.

   6 practice tasks.

3. Differentiate an accumulated quantity

   Lesson details and objectives

   Understand the first fundamental theorem through small added intervals and variable bounds.

   • Connect accumulation derivative to the current integrand value.
   • Use chain factors for variable endpoints.
   • State continuity assumptions and interpret graph behavior.

   6 practice tasks.

4. Recover a function and evaluate a definite integral

   Lesson details and objectives

   Separate a family of primitives, an initial-value solution and a definite numerical accumulation.

   • Find basic antiderivatives and verify by differentiation.
   • Use initial conditions to choose a constant.
   • Evaluate definite integrals by the fundamental theorem.

   6 practice tasks.

5. Reverse a chain rule without losing the bounds

   Lesson details and objectives

   Transform the integrand, differential and endpoints consistently.

   • Choose and justify a substitution.
   • Convert definite bounds or back-substitute consistently.
   • Recognize missing factors and unsuitable substitutions.

   6 practice tasks.

6. Find a region before integrating its height

   Lesson details and objectives

   Use intersections, orientation and splitting to construct nonnegative area.

   • Identify top-minus-bottom or right-minus-left.
   • Split where boundary order changes.
   • Distinguish signed differences from geometric area.

   6 practice tasks.

7. Accumulate signed velocity and total travel

   Lesson details and objectives

   Recover position while separating displacement, distance and observations.

   • Integrate velocity with an initial position.
   • Split speed at velocity zeros.
   • Interpret acceleration data without inventing initial conditions.

   6 practice tasks.

8. Build volume from cross-sectional area

   Lesson details and objectives

   Derive disk and washer integrands from distances to a named rotation axis.

   • Choose outer and inner radii as distances.
   • Use cross-sectional area times thickness.
   • Handle shifted axes and distinguish holes from negative radii.

   6 practice tasks.

9. Count thin cylindrical layers

   Lesson details and objectives

   Choose a slicing direction that avoids unnecessary inverse branches.

   • Derive the shell factor radius times circumference.
   • Construct shell height from a region.
   • Avoid double counting when a region crosses the axis.

   6 practice tasks.

10. Replace a varying function by an equal-total constant

    Lesson details and objectives

    Use interval normalization and distinguish average value from endpoint averages.

    • Derive integral average value and its units.
    • Apply the mean value theorem for integrals under continuity.
    • Interpret averages with the correct weighting.

    6 practice tasks.

11. Reverse the product rule strategically

    Lesson details and objectives

    Choose parts that simplify a problem and handle endpoint terms explicitly.

    • Derive integration by parts from the product rule.
    • Evaluate definite and repeated-parts integrals.
    • Recognize cyclic integrals and verify symbolic results.

    6 practice tasks.

12. Split a rational function into integrable pieces

    Lesson details and objectives

    Use division, factor structure and coefficient identities without losing singularities.

    • Divide before decomposing an improper rational expression.
    • Include repeated-factor and irreducible-quadratic terms.
    • Integrate logarithmic and rational pieces on valid intervals.

    6 practice tasks.

13. Use identities to expose an inner derivative

    Lesson details and objectives

    Choose odd-power substitutions or half-angle identities in radian trigonometry.

    • Integrate products of sine and cosine by parity.
    • Use power-reduction identities for even powers.
    • Handle tangent/secant products with domain restrictions.

    6 practice tasks.

14. Choose a branch before simplifying a radical

    Lesson details and objectives

    Use trigonometric identities to simplify quadratic radicals without dropping absolute values.

    • Match quadratic radical forms with substitutions.
    • Choose angle ranges and track differentials.
    • Return to the original variable and verify branches.

    6 practice tasks.

15. Define each infinite or singular contribution by a limit

    Lesson details and objectives

    Separate convergent improper integrals from misleading cancellations.

    • Replace infinite endpoints and singularities with limits.
    • Use p-integral and comparison tests.
    • Distinguish ordinary improper convergence from principal value.

    6 practice tasks.

16. Estimate an integral and justify an error bound

    Lesson details and objectives

    Compare rules, refinement evidence and derivative-based guarantees.

    • Compute composite midpoint, trapezoid and Simpson rules.
    • Apply error bounds only with appropriate derivative bounds and partitions.
    • Separate discretization error from data and model errors.

    6 practice tasks.