The purpose of this research is to consider students’ mathematical ideas of how the mensuration formula is deduced and how these ideas are applied in the process of first revising how to deduce the area of a circle and then trying to deduce the volume of a sphere through the teaching unit of integral calculus. In this unit students express the process of finding the volume of a sphere mathematically and interpret it by reviewing the process of obtaining the area of a circle. Therefore we design and practice the unit of integral calculus in order to deduct mensuration formula for circle and sphere. We consider typical student’s activities by the qualitative method in the teaching unit of integral calculus. Then we clarify students’ mathematical ideas and how the students apply these ideas effectively.
Students’ activities through classes can be summarized into four points. First, they expressed the description of the arithmetic textbook mathematically, which helped them interpret the area formula of a circle more deeply and give new ideas for deducing the volume formula of a sphere. Secondly, students interpreted the process of deducing the volume formula of a sphere by connecting mathematical ideas in reproducing the area formula of a circle with transition between two dimensions and three dimensions. Third, an encounter with a circular argument made the students recognize that they should always bear precondition in mind. At the same time, it gave them a good opportunity to explain about their ideas to others. Finally, students attempted to apply the ideas of the area formula of a circle and the volume formula of a sphere which were produced in the previous stage by modifying and improving their ideas.