To find the area between two curves, you must first find the points of intersection. Setting f(x) equal to g(x), You have:
√x-7 = x-7
Squaring both sides, You get:
x-7 = x^2 - 14x + 49
Simplifying and rearranging, you get:
x^2 - 15x + 56 = 0
Factoring, we get:
(x-7)(x-8) = 0
So x = 7 or x = 8.
Now we can find the area between the curves by integrating from x = 7 to x = 8:
∫[7,8] (f(x) - g(x)) dx
= ∫[7,8] (√x-7 - (x-7)) dx
= ∫[7,8] (√x - x) dx
I can simplify this integral by using u-substitution. Let u = √x and du = 1/(2√x) dx. Then:
∫[7,8] (√x - x) dx
= ∫[√7,√8] (u^2 - u^2) du (since √7=7^0.5 and √8=8^0.5)
= 0
Therefore, the area between the curves is 0.