Let f and g be continuous functions with the following properties.?

Question

Let f and g be continuous functions with the following properties.
i. g(x)=A-f(x) where A is a constant
ii. integ(1,2)f(x)dx=integ(2,3)g(x)dx
iii. -3A=integ(2,3)g(x)dx
a. find integ(1,3)f(x)dx in terms of A
b. find the average value of g(x) in terms of A, over the interval [1,3]
c. find the value of k if integ.(0,1)f(x+1)dx=kA

Answer 1

a) [1, 3]∫f(x) dx
= [1, 2]∫f(x) dx + [2, 3]∫f(x) dx
= [2, 3]∫g(x) dx + [2, 3]∫f(x) dx (from ii)
= [2, 3]∫g(x) dx + [2, 3]∫A - g(x) dx (from i, since f(x) = A-g(x)
= [2, 3]∫A dx
= A

b) The average value of g(x) over [1, 3] is:
1/2 [1, 3]∫g(x) dx
1/2 [1, 3]∫A - f(x) dx
1/2 ([1, 3]∫A dx - [1, 3]∫f(x) dx)
1/2 (2A - A) (from part a)
A/2

c) [0, 1]∫f(x+1) dx = kA
[1, 2]∫f(x) dx = kA
[2, 3]∫g(x) dx = kA (by ii)
-3A = kA (by iii)
k = -3