Accumulation and definite integrals
What students will learn
Connect signed rate sums, numerical approximations, and accumulation functions through the Fundamental Theorem.
Before this topic
- Rates and units
- Derivative rules and graph signs
Read and connect
Read: Exploring Accumulations of Change →
Accumulate signed rate contributions and distinguish net change from total activity and final amount.
Read: Approximating Areas with Riemann Sums →
Use each interval width in rectangular or trapezoidal approximations and justify any error direction.
Read: Riemann Sums, Summation Notation, and Definite Integral Notation →
Translate partitions and sampled inputs between Riemann sums and definite integral notation.
Read: The Fundamental Theorem of Calculus and Accumulation Functions →
Differentiate an accumulation function using continuity, moving endpoints, and the chain rule.
Read: Interpreting the Behavior of Accumulation Functions Involving Area →
Use the integrand signs and trends to analyze an accumulation function and compare its candidate values.
Read: Applying Properties of Definite Integrals →
Use interval addition, reversal, linearity, symmetry, and bounded discontinuity properties before calculating.
Read: The Fundamental Theorem of Calculus and Definite Integrals →
Check an antiderivative and continuity, then evaluate signed accumulation as an endpoint difference.
Check understanding
Use the complete prompt to make student reasoning visible, then consult the separate worked reasoning and criteria. A reading completion tick records reading history.
Weight an unequal partition correctly →
Compute left, right, and trapezoidal sums with each width and explain why an unweighted sample mean gives a different result.
This table-based task does not assess the full graphical, analytic, and verbal range of integral approximations.
Differentiate a moving-endpoint accumulation →
Apply continuity, the Fundamental Theorem, and the endpoint’s chain factor without evaluating the original integral.
The integral is supplied; this checks differentiation rather than constructing an integral representation.
Construct and sketch accumulation from a rate graph →
Build net-change and amount integrals, use signed areas and endpoint comparisons, then justify and draw the accumulation curve.
Uses the supplied graph; numerical, analytic, and verbal construction contexts require separate checks.
Standards and earlier reading lists
These topic placements organize reading. Edition-qualified Standards mappings and the existing complete unit task sets retain their reviewed scope.