For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 135. x=(1+y^2)/y,y=1,y=4, and the y-axis

To solve this problem using the method of cylindrical shells, we need to find the volume generated by rotating the region bounded by the given curve, \( y = 1 \), \( y = 4 \), and the y-axis about the x-axis.

Given the curve \( x = \frac{1 + y^2}{y} \), the limits are from \( y = 1 \) to \( y = 4 \), and we rotate around the x-axis. Using the method of cylindrical shells, the formula for the volume is:

\[
V = 2\pi \int_{y_1}^{y_2} r(y) h(y) \, dy
\]

Where:
- \( r(y) \) is the radius of the shell, which is just \( y \) (the distance from the x-axis).
- \( h(y) \) is the height of the shell, which is the expression for \( x \), given by \( h(y) = \frac{1 + y^2}{y} \).

Now, we set up the integral:

\[
V = 2\pi \int_{1}^{4} y \cdot \frac{1 + y^2}{y} \, dy
\]

Simplifying the integrand:

\[
V = 2\pi \int_{1}^{4} (1 + y^2) \, dy
\]

Now, we compute the integral:

\[
V = 2\pi \left[ y + \frac{y^3}{3} \right]_{1}^{4}
\]

Evaluating at the limits:

At \( y = 4 \):

\[
4 + \frac{4^3}{3} = 4 + \frac{64}{3} = 4 + 21.33 = 25.33
\]

At \( y = 1 \):

\[
1 + \frac{1^3}{3} = 1 + \frac{1}{3} = 1.33
\]

Subtract the two results:

\[
25.33 - 1.33 = 24
\]

Finally, multiply by \( 2\pi \):

\[
V = 2\pi \times 24 = 48\pi
\]

Thus, the volume of the solid generated by rotating the region about the x-axis is \( 48\pi \).

---

Would you like further clarification on any step?

Here are some related problems to try:

1. Find the volume of the region between \( x = 1 + y^2 \) and \( y = 0 \), rotated about the x-axis.
2. Use cylindrical shells to find the volume of the solid obtained by rotating \( y = x^2 \), \( y = 0 \), and \( x = 1 \) around the y-axis.
3. Calculate the volume of the region bounded by \( y = \sin(x) \) and rotated about the y-axis.
4. Find the volume of the solid formed by rotating the region between \( y = e^x \), \( y = 0 \), and \( x = 1 \) around the x-axis.
5. Compute the volume of the region under \( y = \frac{1}{x} \) between \( x = 1 \) and \( x = 4 \) rotated about the x-axis.

**Tip:** Always visualize the problem before integrating to set up the limits and the function correctly!

For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 135. x=(1+y^2)/y, y=1, y=4, and the y-axis.

Learn how to find the volume of a solid generated by rotating the region between the curve x=(1+y^2)/y, y=1, y=4, and the y-axis around the x-axis. This detailed solution uses the cylindrical shells method, covering key calculus concepts like integration and volumes of revolution. Ideal for undergraduate calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation