一招解决积分学中的“偶倍奇零”问题

One Way to Solve the Problem of “Even Times and Odd Zeros” in Calculus

摘要: 微积分中，定积分，二重积分，三重积分，第一类线面积分经常碰到积分区域区域有对称性，被积函数具有奇偶性的状况。如果针对每个积分列举偶倍奇零公式，则其繁杂程度超出学生的接受程度。如果要求学生根据这些公式来做题，则背离了高等教育的目的。特别对非数学专业的学生，微积分的教学应该提供给他们更为直观的解决方案！将所有的无方向的积分看作(广义)质量，则简单高效地解决了所有“偶倍奇零”问题。同时解决类似的含有对称性的积分问题！

Abstract: In calculus, the situation that the integration region has symmetry and the integrand has parity occur a lot in definite integrals, double integrals, triple integrals, line integrals of the first type, and area integrals of the first type, etc. If we list even-multiple-odd-zero formulas for each integral, the complexity is beyond the acceptance of students. If students are required to solve the problems by means of these formulas, it deviates from the purpose of higher education. The teaching of calculus should provide them with more intuitive solutions, especially for non-mathematics majors! We can solve all “even-multiple-odd-zero” problems simply and efficiently by treating all undirected integrals as (generalized) masses. At the same time, similar integration problems with symmetry can also be solved!