Find the area bounded by the courves y=1/x, y=1/x^2 and x=8

To find the area bounded by the curves \( y = \frac{1}{x} \), \( y = \frac{1}{x^2} \), and the vertical line \( x = 8 \), we first need to determine the region of integration.

### Step 1: Determine points of intersection
We want to find the \(x\)-coordinates where the curves \( y = \frac{1}{x} \) and \( y = \frac{1}{x^2} \) intersect. Set the equations equal to each other:
\[
\frac{1}{x} = \frac{1}{x^2}
\]
Multiply both sides by \(x^2\) to eliminate the denominators:
\[
x = 1
\]
Thus, the curves intersect at \(x = 1\). Therefore, the area we need to calculate is between \(x = 1\) and \(x = 8\).

### Step 2: Set up the integral
The area between the curves can be computed as the integral of the difference between the two functions:
\[
A = \int_{1}^{8} \left( \frac{1}{x} - \frac{1}{x^2} \right) dx
\]

### Step 3: Compute the integral
Let's compute the integral step by step:
\[
A = \int_{1}^{8} \left( \frac{1}{x} - \frac{1}{x^2} \right) dx
\]
We can split the integral:
\[
A = \int_{1}^{8} \frac{1}{x} dx - \int_{1}^{8} \frac{1}{x^2} dx
\]

#### First integral:
\[
\int \frac{1}{x} dx = \ln|x|
\]
Evaluating from 1 to 8:
\[
\left[ \ln|x| \right]_{1}^{8} = \ln(8) - \ln(1) = \ln(8)
\]

#### Second integral:
\[
\int \frac{1}{x^2} dx = -\frac{1}{x}
\]
Evaluating from 1 to 8:
\[
\left[ -\frac{1}{x} \right]_{1}^{8} = -\frac{1}{8} + 1 = \frac{7}{8}
\]

### Step 4: Calculate the total area
Now, subtract the results of the two integrals:
\[
A = \ln(8) - \frac{7}{8}
\]
We know that \( \ln(8) = 3\ln(2) \approx 2.0794 \), so:
\[
A \approx 2.0794 - 0.875 = 1.2044
\]

Thus, the area bounded by the curves is approximately \( 1.2044 \) square units.

Would you like more details on this solution or have any further questions?

Here are some related questions:
1. What is the area between \( y = \frac{1}{x^2} \) and the x-axis between \( x = 1 \) and \( x = 8 \)?
2. How does changing the boundary \( x = 8 \) to another value affect the area?
3. What is the general formula for the area between two curves?
4. How would this area change if one of the curves was shifted vertically?
5. How would you find the volume if this region was rotated around the x-axis?

**Tip:** To simplify solving problems involving areas between curves, always plot the curves first to visualize the region you are integrating.

Find the area bounded by the curves y = 1/x, y = 1/x^2 and x = 8

This page provides a detailed solution to finding the area bounded by the curves y = 1/x, y = 1/x^2, and the vertical line x = 8. The solution walks through key calculus concepts, including the use of definite integrals and the area between curves. Suitable for advanced high school or introductory university students.

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