y=2/x^2 ,x=1, x=0 find volume using shell method and revolve around x axis 

To find the volume of the solid formed by revolving the curve \( y = \frac{2}{x^2} \) around the **x-axis** between \( x = 1 \) and \( x = 0 \) using the **shell method**, we will follow these steps:

### Step 1: Set up the shell method formula
The volume \( V \) of a solid of revolution using the shell method is given by:

\[
V = 2\pi \int_{a}^{b} (radius)(height) \, dx
\]

For this problem:
- The **radius** is the distance from the x-axis to the function, which is \( y = \frac{2}{x^2} \).
- The **height** is the x-coordinate, \( x \).

Thus, the formula becomes:

\[
V = 2\pi \int_{0}^{1} x \cdot \frac{2}{x^2} \, dx
\]

### Step 2: Simplify the integrand
Simplify the expression inside the integral:

\[
x \cdot \frac{2}{x^2} = \frac{2x}{x^2} = \frac{2}{x}
\]

### Step 3: Set up the integral
Now we have the following integral to compute:

\[
V = 2\pi \int_{0}^{1} \frac{2}{x} \, dx
\]

### Step 4: Solve the integral
The integral of \( \frac{1}{x} \) is \( \ln |x| \). So, we compute:

\[
V = 2\pi \left[ 2 \ln |x| \right]_{0}^{1}
\]

### Step 5: Evaluate the limits
Now evaluate the definite integral:

\[
V = 2\pi \left( 2 \ln(1) - 2 \ln(0) \right)
\]

- \( \ln(1) = 0 \)
- \( \ln(0) \) approaches negative infinity, so this volume diverges (it goes to infinity).

### Conclusion:
The volume of the solid is **infinite**, as the integral does not converge due to the behavior at \( x = 0 \). The function grows without bound near this point.

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Would you like further clarification on any part of this solution?

Here are 5 related questions you might consider:
1. What conditions lead to a finite volume in the shell method?
2. How would the result change if we revolved around a different axis?
3. Can the disk or washer method be used for this problem?
4. What happens if we set different bounds, excluding \( x = 0 \)?
5. How does the function \( y = \frac{2}{x^2} \) behave as \( x \) approaches 0?

**Tip:** When dealing with improper integrals, always check if the function has any singularities (points where it becomes infinite). This helps determine if the volume will be finite or infinite.

y=2/x^2, x=1, x=0 find volume using shell method and revolve around x-axis

Learn how to calculate the volume of a solid formed by revolving the curve y = 2/x^2 around the x-axis using the shell method. This calculus problem involves improper integrals and the shell method formula. Suitable for undergraduate students studying calculus.

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