Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Finding an expression for the length of a curve is one of the simpler geometric applications of the integral. If f is a function with a continuous derivative, then the expression gives the length of the curve y = f (x) on an interval [a, b]. However, after writing out the integrand for familiar functions such as y = x2 and y = sin x, it quickly becomes apparent that, in general, finding an antiderivative is a challenge. Of course, a computer can give accurate approximations for the value of the integral for the length of a curve, but it would be nice to find the exact length rather than a decimal approximation. In his work on geometry (from 1637), Descartes stated that he believed it was not possible to determine the exact lengths of curves. However, just twenty years later, William Neile was able to find the length of arcs of semicubical parabolas (see Katz [1]). These curves have the form y = kx3/2 and are usually the first examples or exercises given to students since the resulting integral is very easy to compute. In this paper, we are going to examine this curve and other related curves and consider problems such as the following: find rational numbers a and b so that the length of the curve over the interval is an [a, b] integer. As we shall see, problems such as this provide a variety of opportunities for undergraduate students to explore some interesting mathematics arising from a few simple and accessible questions.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.005 | 0.005 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it