Math Snap

STEP 1

1. The functions $y = \frac{1}{5x}$ and $y = \frac{1}{x^2}$ intersect at some point, defining the limits of integration.
2. The vertical line $x = 11$ is the upper boundary for the region of interest.
3. The area is calculated between the two functions from the point of intersection to $x = 11$.

STEP 2

1. Find the intersection point of the two functions.
2. Determine the limits of integration.
3. Set up the integral for the area.
4. Evaluate the integral to find the exact area.

STEP 3

Find the intersection point of the two functions by setting them equal:
$$\frac{1}{5x} = \frac{1}{x^2}$$ Solve for $x$:
$$x^2 = 5x$$ $$x(x - 5) = 0$$ The solutions are $x = 0$ and $x = 5$.

STEP 4

Determine the limits of integration. The region of interest is between $x = 5$ and $x = 11$.
Lower limit $a = 5$.
Upper limit $b = 11$.

STEP 5

Set up the integral for the area. The function $g(x)$ is the difference between the two functions:
$$g(x) = \frac{1}{5x} - \frac{1}{x^2}$$ The integral is:
$$A = \int_{5}^{11} \left( \frac{1}{5x} - \frac{1}{x^2} \right) \, dx$$

Evaluate the integral:
$$A = \int_{5}^{11} \left( \frac{1}{5x} - \frac{1}{x^2} \right) \, dx$$ Calculate each part separately:
$$\int \frac{1}{5x} \, dx = \frac{1}{5} \ln |x|$$ $$\int \frac{1}{x^2} \, dx = -\frac{1}{x}$$ Thus, the integral becomes:
$$A = \left[ \frac{1}{5} \ln |x| + \frac{1}{x} \right]_{5}^{11}$$ Evaluate at the bounds:
$$A = \left( \frac{1}{5} \ln 11 + \frac{1}{11} \right) - \left( \frac{1}{5} \ln 5 + \frac{1}{5} \right)$$ Simplify:
$$A = \frac{1}{5} (\ln 11 - \ln 5) + \frac{1}{11} - \frac{1}{5}$$ $$A = \frac{1}{5} \ln \left( \frac{11}{5} \right) + \frac{1}{11} - \frac{1}{5}$$ The exact area of the region is:
$$\boxed{\frac{1}{5} \ln \left( \frac{11}{5} \right) + \frac{1}{11} - \frac{1}{5}}$$