Prove that the areas bounded by the -axis and half-waves of the curve

MATHEMATICS · Subjective · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Prove that the areas

S_0, S_1, S_2, \dots

bounded by the

X

-axis and half-waves of the curve

y = e^{-\alpha x} \sin \beta x, x \ge 0

. form a geometric progression with the common ratio

g = e^{-\pi \alpha / \beta}