Question

Decide without calculation if each of the integrals below are positive, negative, or zero?

1) Positive
2) Negative
3) Zero

Answer 1

When evaluating the integral, we consider the range of integration and the value of the function. The integral is zero outside the range where the particle exists and is positive within the range zero to L if the function is positive there.

Step-by-step explanation:

The evaluation of an integral depends on both the function being integrated and the limits of integration. Based on the information given, we can deduce that the particle is constrained within a tube, hence the function C is zero outside the tube. Considering the integral over three parts, we'll apply these principles:

1. Negative infinity to zero: The particle cannot exist here, so the function C is zero, and the integral will also be zero.
2. Zero to L: The particle exists within this range, and if the function C is positive within the tube, the integral would be positive.
3. L to infinity: Similar to the first part, the particle cannot exist here, so the function C is zero, making the integral zero.

Therefore, the first and last integrals are zero and do not contribute to the overall integral's sign. Only the integration from zero to L would contribute, and it would be positive if the function C is positive within that interval.