Problem 15 - College Algebra and Calculus: An Applied Approach

Question

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ y=\frac{1}{x^{2}}, y=0, x=1, x=5 $$

Step-by-Step Solution

Answer

The area of the region bounded by the given functions is \(\frac{4}{5}\) square units.

Step 1: Understand the Functions

The given equations are \(y=\frac{1}{x^{2}}\), \(y=0\), \(x=1\), and \(x=5\). The first equation represents a hyperbola that opens downward and upward. The second equation represents the x-axis, and the final two equations are vertical lines. These four equations together form a closed region.

Step 2: Set up the Integral

The area A of the region under a non-negative curve from a to b is given by the integral\[A = \int_{a}^{b}f(x) dx\]In this case, the function \(f(x) = \frac{1}{x^{2}}\), and the region goes from \(x = 1\) to \(x = 5\). So, plug these into the equation for A:\[A = \int_{1}^{5} \frac{1}{x^{2}} dx\]

Step 3: Evaluate the Integral

Now we evaluate the integral using integral calculus. The antiderivative of \(\frac{1}{x^{2}}\) is \(-\frac{1}{x}\). Using the fundamental theorem of calculus, we find\[A =-\frac{1}{x} \biggr|_{1}^{5}\]Evaluating this gives\[A = -\frac{1}{5} -(-\frac{1}{1}) = \frac{4}{5}\]