Find the volume of the solid obtained by rotating the region bounded by $y = \frac{1}{x}$, $x = 2$, $x = 5$, and $y = 0$ about the $y$-axis.

Neetesh Kumar | November 09, 2024

Calculus Homework Help

This is the solution to DHW Calculus
Assignment: 2 Question Number 15
Contact me if you need help with Homework, Assignments, Tutoring Sessions, or Exams for STEM subjects.
You can see our Testimonials or Vouches from here of the previous works I have done.

Get Homework Help

---

Step-by-step solution:

Step 1: Volume formula for rotation about the $y$-axis

To find the volume of the solid formed by rotating a region about the $y$-axis, we use the method of cylindrical shells. The volume $V$ is given by the formula:

$V = 2\pi \int_{a}^{b} \text{(radius)} \times \text{(height)} \, dx$

Here:

• The radius of each shell is the distance from the $y$-axis, which is simply the $x$-coordinate, so the radius is $x$.
• The height of each shell is the value of the function $y = \frac{1}{x}$, which gives the height of the shell at each $x$.
• The limits of integration are from $x = 2$ to $x = 5$ (the bounds of the region).

Step 2: Set up the integral

The volume integral becomes:

$V = 2\pi \int_{2}^{5} x \cdot \frac{1}{x} \, dx$

Simplify the integrand:

$V = 2\pi \int_{2}^{5} 1 \, dx$

Step 3: Evaluate the integral

Now, evaluate the integral:

$V = 2\pi \left[ x \right]_{2}^{5}$

Substitute the limits of integration:

$V = 2\pi \left[ 5 - 2 \right]$

$V = 2\pi \times 3 = 6\pi$

Final Answer:

The volume of the solid is:

$V = 6\pi \approx 18.8496 \, \text{cubic units}$

Thus, the volume is approximately 18.85 cubic units.

Please comment below if you find any error in this solution.
If this solution helps, then please share this with your friends.
Please subscribe to my Youtube channel for video solutions to similar questions.
Keep Smiling :-)