Questions: 1.) Sketch the region enclosed by the curves below. 2.) Decide whether to integrate with respect to x or y. 3.) Find the area of the region. y=1/x, y=1/x^2, x=2 Area =

Solution Steps

The curves given are: \[ y = \frac{1}{x} \] \[ y = \frac{1}{x^2} \] \[ x = 2 \]

To find the points of intersection between \( y = \frac{1}{x} \) and \( y = \frac{1}{x^2} \), set the equations equal to each other: \[ \frac{1}{x} = \frac{1}{x^2} \] \[ x = 1 \]

Since the functions are given in terms of \( y \) as functions of \( x \), it is easier to integrate with respect to \( x \).

The area \( A \) is given by the integral: \[ A = \int_{1}^{2} \left( \frac{1}{x} - \frac{1}{x^2} \right) \, dx \]

\[ \int \frac{1}{x} \, dx = \ln|x| \] \[ \int \frac{1}{x^2} \, dx = -\frac{1}{x} \]

Evaluating the definite integral from 1 to 2: \[ A = \left[ \ln|x| + \frac{1}{x} \right]_{1}^{2} \] \[ A = \left( \ln 2 + \frac{1}{2} \right) - \left( \ln 1 + 1 \right) \] \[ A = \ln 2 + 0.5 - 1 \] \[ A = \ln 2 - 0.5 \]

Final Answer