Let S be the solid obtained by rotating the region shown in the figure below about the y-axis.
y=4sin(2x^2), x=sqrt(pi/2). Use the method of cylindrical shells to find the volume V of S.What are the circumference c and height h of a typical cylindrical shell?

To solve this problem using the method of cylindrical shells, let's first break it down into the key components and then answer the specific question about the circumference \( c \) and height \( h \) of a typical shell.

### Volume of Solid Using Cylindrical Shells:
The formula for the volume \( V \) of a solid of revolution obtained by rotating a region around the y-axis using cylindrical shells is:

\[
V = \int_{a}^{b} 2\pi x \cdot h(x) \, dx
\]

Where:
- \( x \) is the radius of the shell (distance from the y-axis),
- \( h(x) \) is the height of the shell (the function defining the region),
- \( 2\pi x \) is the circumference of the shell,
- \( [a, b] \) is the interval for \( x \).

### Step 1: Circumference \( c \)
For the method of cylindrical shells, the **radius** of a typical shell is the \( x \)-coordinate of the point on the curve being rotated. The **circumference** of the shell is the distance around the shell, given by:

\[
c = 2\pi x
\]

Thus, the circumference of a typical cylindrical shell at \( x \) is:

\[
c = 2\pi x
\]

### Step 2: Height \( h \)
The height of a typical shell is the vertical distance between the curve \( y = 4\sin(2x^2) \) and the x-axis (since we're rotating around the y-axis). Therefore, the height of the shell is simply the value of the function at that point, i.e., \( h(x) = 4\sin(2x^2) \).

So, the **height** of a typical shell is:

\[
h(x) = 4\sin(2x^2)
\]

### Step 3: Interval of Integration
Since we are given that \( x = \sqrt{\frac{\pi}{2}} \) is the upper limit of \( x \), the volume integral will be evaluated over the interval \( [0, \sqrt{\frac{\pi}{2}}] \).

Let me know if you'd like to proceed with calculating the full volume or if you have any questions about these steps.

Here are 5 questions that extend the current problem:

1. How do you compute the integral of \( 4 \sin(2x^2) \)?
2. What happens to the method of cylindrical shells when rotating around an axis other than the y-axis?
3. Can you use the disk method to solve this problem instead of cylindrical shells?
4. How do you determine the limits of integration when rotating around the x-axis?
5. What role does symmetry play in calculating volumes of solids of revolution?

**Tip:** Always identify the axis of rotation clearly before applying any integration method.

Let S be the solid obtained by rotating the region shown in the figure below about the y-axis. y=4sin(2x^2), x=sqrt(pi/2). Use the method of cylindrical shells to find the volume V of S. What are the circumference c and height h of a typical cylindrical shell?

Solve for the volume of a solid obtained by rotating the region y=4sin(2x^2) about the y-axis using the method of cylindrical shells. Learn how to calculate the circumference and height of a typical shell in this detailed calculus problem. Suitable for undergraduate 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