Step-by-step explanation:
Let [tex]f(x) = 4[/tex] and [tex]g(x) = \frac{1}{x^2}[/tex]. The area A of the region bounded by the given lines is
[tex]\displaystyle A = \int [f(x) - g(x)]dx[/tex]
Note that [tex]g(x) = \frac{1}{x^2}[/tex] intersects y = 4 at x = 1/2 so the limits of integration go from x = 1/2 to x = 5. The area integral can then be written as
[tex]\displaystyle A = \int_{\frac{1}{2}}^{5}\left(4 - \dfrac{1}{x^2}\right)dx[/tex]
[tex]\:\:\:\:= \left(4x + \dfrac{1}{x}\right)_{\frac{1}{2}}^5[/tex]
[tex]\:\:\:\:= (20 + \frac{1}{5}) - (2 + 2) = \dfrac{81}{5} = 16\frac{1}{5}[/tex]
