342
6.1. USING DEFINITE INTEGRALS TO FIND AREA AND LENGTH
(a) Find the average value of f (x) = 2 − x 2 on the interval [0,
√
2]. Call this value
r.
(b) Sketch a graph of y = f (x) and y = r. Find their intersection point(s).
(c) Show that on the interval [0,
√
2], the amount of area that lies below y = f (x)
and above y = r is equal to the amount of area that lies below y = r and above
y = f (x).
(d) Will the result of (c) be true for any continuous function and its average value
on any interval? Why?