True or false. If true, why? If false, what is a counter example, and can you explain why it contradicts? Please

Question

If f and g continuous function on [a,b] then (upper: b, lower a)∫f(x)g(x)dx=[(upper: b, lower a)∫f(x)dx]*[(upper: b, lower a)∫g(x)dx].

Answer 1

FALSE:

 

Let f(x) = x, g(x) = x² where 0 ≤ x ≤ 1

 

Then ∫f(x)dx from x = 0 to x = 1 is 1/2

       

        ∫g(x)dx from x = 0 to x = 1 is 1/3

 

        f(x)g(x) = x³  So, ∫[f(x)g(x) from x = 0 to x = 1 is 1/4

 

But,  ∫f(x)dx ∫g(x)dx from x = 0 to x = 1 is (1/2)(1/3) = 1/6

 

So, ∫[f(x)g(x)]dx ≠ ∫f(x)dx ∫g(x)dx