### Finding the Volume of a Solid Using Cylindrical Shells**Problem Statement:**Use the method of cylindrical shells to find the volume of the solid generated by rotating the region bounded by $ y = \cos \left( \frac{\pi x}{2} \right) $, $ y = 0 $, and $ 0 \leq x \leq 1 $ about the $ y $-axis. State the integration method used to evaluate the integral.---**Detailed Explanation:**To find the volume of the solid using the method of cylindrical shells, follow these steps:1. **Identify the Function and Bounds:** - The function provided is $ y = \cos \left( \frac{\pi x}{2} \right) $. - The region is bounded by $ y = 0 $ and $ 0 \leq x \leq 1 $.2. **Setup for Cylindrical Shells:** The formula for the volume $ V $ using the method of cylindrical shells when rotating around the y-axis is given by: \[ V = \int_{a}^{b} 2\pi x \cdot f(x) \, dx \] Here, $ f(x) = \cos \left( \frac{\pi x}{2} \right) $ and the bounds are $ a = 0 $ and $ b = 1 $.3. **Formulating the Integral:** \[ V = \int_{0}^{1} 2\pi x \cdot \cos \left( \frac{\pi x}{2} \right) \, dx \]4. **Integration Method:** To evaluate this integral, we will likely need to use integration by parts or a suitable substitution.**Step-by-Step Solution:**1. **Integration by Parts:** Let's use integration by parts, where: \[ \int u \, dv = uv - \int v \, du \] Choose $ u $ and $ dv $ as follows: \[ u = x \quad \text{and} \quad dv = 2\pi \cos \left( \frac{\pi x}{2} \right) dx \]2. **Derivatives and Antiderivatives:**

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Use the method of cylindrical shells to find the volume of the solid generated by rotating the region bounded by y = cos =), y = 0, and0 < x<1 about the y-axis. State the 2 integration method used to evaluate the integral.

Use the method of cylindrical shells to find the volume of the solid generated by rotating the region bounded by y = cos =), y = 0, and0 < x<1 about the y-axis. State the 2 integration method used to evaluate the integral.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

### Finding the Volume of a Solid Using Cylindrical Shells

**Problem Statement:**

Use the method of cylindrical shells to find the volume of the solid generated by rotating the region bounded by $ y = \cos \left( \frac{\pi x}{2} \right) $, $ y = 0 $, and $ 0 \leq x \leq 1 $ about the $ y $-axis. State the integration method used to evaluate the integral.

---

**Detailed Explanation:**

To find the volume of the solid using the method of cylindrical shells, follow these steps:

1. **Identify the Function and Bounds:**
- The function provided is $ y = \cos \left( \frac{\pi x}{2} \right) $.
- The region is bounded by $ y = 0 $ and $ 0 \leq x \leq 1 $.

2. **Setup for Cylindrical Shells:**
The formula for the volume $ V $ using the method of cylindrical shells when rotating around the y-axis is given by:
\[
V = \int_{a}^{b} 2\pi x \cdot f(x) \, dx
\]
Here, $ f(x) = \cos \left( \frac{\pi x}{2} \right) $ and the bounds are $ a = 0 $ and $ b = 1 $.

3. **Formulating the Integral:**
\[
V = \int_{0}^{1} 2\pi x \cdot \cos \left( \frac{\pi x}{2} \right) \, dx
\]

4. **Integration Method:**
To evaluate this integral, we will likely need to use integration by parts or a suitable substitution.

**Step-by-Step Solution:**

1. **Integration by Parts:**
Let's use integration by parts, where:
\[
\int u \, dv = uv - \int v \, du
\]
Choose $ u $ and $ dv $ as follows:
\[
u = x \quad \text{and} \quad dv = 2\pi \cos \left( \frac{\pi x}{2} \right) dx
\]

2. **Derivatives and Antiderivatives:**

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