From Example 4 of Section 2.3 of Apostol, we know that given a nonnegative, integrable function on an interval , and the area of the ordinate set of is , then if we define a function , we have,
Generalize this formula and prove the result.
To generalize this, we proceed as follows.
Let be a nonnegative integrable function on , and let be the ordinate set of . If we apply a transformation under which we multiply the -coordinate of each point on the graph of by a constant and each -coordinate by a constant , then we obtain a new function were a point is on if and only if is on . Then,
Letting denote the ordinate set of :