Characterizing Power Functions by Volumes of Revolution

Bettina Richmond, Tom Richmond
1998 The College Mathematics Journal  
A power function is a function of the form f (x) = x n . We will consider positive multiples of power functions with positive powers, that is, functions of the form f (x) = kx n (k, n > 0) over the domain {x : x ≥ 0}. For r > 0, let R 1 (r) be the first quadrant region under the curve y = f (x) over an interval [0, r], and let R 2 (r) be the first quadrant region to the left of R 1 (r), as shown in Figure 1 below. Revolve these regions R 1 (r) and R 2 (r) around the y-axis to get solids of
more » ... get solids of revolution with volumes V 1 (r) and V 2 (r), respectively. We will show that the ratio of the volumes V 1 (r) and V 2 (r) is constant, and that this property can be used to characterize these power functions. Figure 1 1
doi:10.2307/2687636 fatcat:zn74pybtyfbf7knhir4xdfkkrm