If, for all real number x, f(x) = g(x) + 5, then on any interval [a, b], what is the area of the region between the graphs of f and g?

∫f(x)dx over [a,b] = ∫(g(x)+5)dx over [a,b].

By Addition rule of integrals, ∫(g(x)+5)dx = ∫g(x)dx + ∫5dx.
= ∫g(x)dx + 5(b-a)

To find the area of the region between the graphs of f(x) and g(x) on an interval [a, b], you need to integrate the difference between the two functions over that interval.

Given that f(x) = g(x) + 5, we can rewrite it as g(x) = f(x) - 5. Now we have an expression for g(x) in terms of f(x).

To find the area between the graphs, we need to evaluate the definite integral of the difference between the two functions:

Area = ∫[a, b] (f(x) - g(x)) dx

Substituting g(x) with f(x) - 5, we get:

Area = ∫[a, b] (f(x) - (f(x) - 5)) dx

Simplifying the expression:

Area = ∫[a, b] (5) dx

Now, integrating with respect to x:

Area = 5 ∫[a, b] dx

The integral of dx is simply x, so we have:

Area = 5(x) evaluated from a to b

Evaluating the definite integral:

Area = 5(b - a)

Therefore, the area of the region between the graphs of f(x) and g(x) on the interval [a, b] is given by the formula 5(b - a).