UTSA Problem-Solving Club

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Calculus: Look Before You Integrate

Fall 2026 · 11 problems

Reflection, reciprocal changes of variables, and geometry reveal the structure of definite integrals. Five warm-ups lead to competition problems, including an integral of an unknown function from Putnam 2016 A3.

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What is in it

Problems only — the club works these out together at meetings. Topic and attribution are listed as a starting point.

  1. Three Rational Integrals Algebra Before CalculusEasy
  2. One Reversal, Many Symmetries Change of Variables / Symmetry
  3. Two Fractions ReflectionEasy
  4. A Reciprocal Reversal Change of Variables / Symmetry / Warm-up
  5. A Circle Away from the Origin Geometry / Shifted Symmetry / Solids of Revolution
  6. Is \(\pi = 3.14… = 22/7\)? Algebra and Positive Area / Putnam First PositionPutnam 1968 A1 (part (a))
  7. An Integral over a Half-Line Improper Integrals / Change of Variables / Competition Style
  8. Logarithms and Square Roots Reflection / Putnam First PositionPutnam 1987 B1
  9. A Function Given Indirectly Functional Equations / Competition Style
  10. An Integral of an Unknown Function Functional Equations / Integral Symmetry / Putnam A3Putnam 2016 A3
  11. A Solid Hidden in an Inequality Rotational Symmetry / Putnam First PositionPutnam 2006 A1