Recall we have defined as the area of the unit disk. We have proved (Apostol, Section 2.3, Example 3) that
Use this and the theorems on the property of integrals to compute the following in terms of :
- Using the expansion/contraction property of the integral (Apostol, Theorem 1.19) we compute:
- Using the expansion/contraction property of the integeral (Apostol, Theorem 1.19) we compute:
- Using the linearity properties of the integral, as well as, expansion/contraction and the formula for given above. We also use this exercise (Apostol, Section 1.25, Exercise #26 part (b)) which establishes that the integral from to of an odd function is 0.