IB Mathematics Analysis and Approaches (AA) HL Internal Assessment Example

The title “Integral calculus and modeling” is concise and effectively signals the mathematical approach employed in this exploration.

The research question is clear but could be rephrased to specify the dependent and independent variables explicitly (e.g., volume as a function of precipitation index).

The “Index” lists a Data collection section without presenting any raw data or summary statistics here; adjust the TOC or include a brief dataset overview to align headings with content.

Consider refining the structure of the Index by grouping the annex content separately or integrating it into main sections for smoother navigation.

The discussion of the IB learner profile is engaging but overly long, detracting from the mathematical focus. Condense this narrative to maintain relevance.

The “Lock volume” heading appears without preceding raw precipitation data; move or rename the preceding section to reflect its content accurately.

Explain more fully why the plan view alone suffices for the ship’s submerged volume approximation and discuss the resulting limitations.

The bibliography entries vary in style (some list authors, others not, inconsistent punctuation). Standardize format (APA or MLA) for consistency.

Notation is inconsistent: the region is called both D and R, and the differential is omitted in places. Standardize symbols and include dx, dy in every integral.

The section heading “Mathematical foundation” clearly frames the theoretical background and sets a solid context for the double integral approach.

Before applying the standard form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, define a and b explicitly and explain how they relate to the original hyperbola foci.

Equation [4] omits the integrals for the central section (ii) and part (iii) is left unfinished. Include all three parts explicitly before evaluation.

Graph 1’s x–axis is not in chronological order, which hinders trend analysis; reorder or explain the sorting of data points.

The author’s personal connection to Panama and family history provides strong motivation and demonstrates clear personal engagement.

The discrepancy between the computed ship draft volume and published 101 000 m³ is noted but only attributed to shape simplification; reflect more critically and consider parameter adjustments.

Linking additional volume requirements to real restrictions on transits shows thoughtful critical reflection on the model’s real–world implications.

Consider performing sensitivity analysis on N and P to quantify how changes in these parameters affect water usage predictions.

The function f(x,y)=304.8 is labeled as draft height but 304.8 m is actually the lock length; clarify which dimension is constant in the integrand.

The iterated integral ${\iint }_{R}304.8dA={\int }_0^{33.5}{\int }_0^{12.8}304.8dydx$ is correctly set up for the rectangular prism.

The final numeric evaluation (130 698.24 m³) lacks unit consistency checks and does not acknowledge that the prism model ignores gate volume; discuss this simplification.

The piecewise hyperbola model (equation [2]) uses large, unexplained coefficients. Provide the derivation steps or simplify to improve transparency.

The step “a=77–3=72” appears without justification for subtracting 3; clarify the source of that adjustment in the geometric context.

The calculation of ${\bar{X}}_{bow}=4.33$ m is presented clearly and demonstrates systematic use of averages to set integration limits.

The expansion of the inner integral in equation [4] correctly separates terms and applies Fubini’s theorem to the complex integrand.

Defining N as 2928 monthly transits then multiplying by 12 implies annual rate, but the text labels N as a monthly average. Clarify this unit distinction.

Equation [6] does not specify the domain of P or handling of negative values; define the functional behavior for drought versus surplus explicitly.