Area between f(x) and 2f(x) given area under f(x)

[LadiesMan]

Suppose the area of the region between the graph of a positive continuous function f and the x-axis from x = a to x = b is 4 square units. Find the area between the area between the curves y = f(x) and y = 2f(x) from x = a to x = b.

Attempt:

Since 2 f(x) is greater than f(x) we can call it g(x) and that will be the dominant function.

$$\int (g(x) - f(x)) dx$$

It becomes...
$$G(x) - F(x)$$

What do I do next?

Thanks

 

[Pere Callahan]

You said g(x)=2f(x), so what can you say about the relation between G(x) and F(x) ..?

Note that you want to consider definite integrals

$$\int_a^b{dx(g(x)-f(x))}$$

Can you simplify g(x)-f(x)..?

 

[LadiesMan]

umm yeah i guess that took me off course.

 

[LadiesMan]

so then how would i do it?

 

[Pere Callahan]

Uhm,

g(x)-f(x) = 2f(x)-f(x) =...?

 

[LadiesMan]

Umm that equals f(x)

 

[Pere Callahan]

very good, so
$$\int_a^b{dx(g(x)-f(x))}=\int_a^b{dx(2f(x)-f(x))}=\int_a^b{dx(f(x))}=...=?$$