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Students in physics courses are often asked to compute integrals that are both similar and different compared to the integrals from Calculus courses. We interviewed three students at a mid-sized midwestern university, and asked them to work on integral problems from math and physics contexts and talk through their thinking. We identified five conceptual resources that students activated throughout and across the interviews. Here, we highlight two of the five resources through an example of a student reasoning through a volume integral problem, and their thought process as they attempt to connect different conceptions of integration. We argue that this case study excerpt is representative of some of the hurdles that certain conceptions of integrals may lead to when solving different integral problems in physics despite students’ prior success in math courses. 

This report describes the synthesis of five new colorful 6-aryldibenzo[b,d]pyrylium cation salts, a largely unexplored structural unit. These rare compounds are benzannulated structural derivatives of the well-known flavylium cations found widespread in natural pigments. These new dyes are directly compared to three previously synthesized 6-aryldibenzo[b,d]pyrylium cation salts as well as eight colorful isomeric 9-aryldibenzo[b,e]pyrylium cation, or 9-arylxanthylium, salts. The 9-arylxanthylium unit is commonly found in the biologically important rhodamine and rosamine dyes, yet six of the analogs presented in this study were either previously unreported or not isolated. The visual and spectroscopic properties of all 16 compounds were analyzed as a function of the structural differences between the compounds. All compounds displayed reversible halochromism in organic solution, displaying bright colors under acidic conditions and becoming colorless under basic conditions.