Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the y-axis. y = 2 + x - x², x + y = 2 V =

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Author:James Stewart
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Chapter1: Functions And Models
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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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### Using the Method of Cylindrical Shells

**Problem:**

Use the method of cylindrical shells to find the volume \( V \) generated by rotating the region bounded by the given curves about the \( y \)-axis.

\[
y = 2 + x - x^2, \quad x + y = 2
\]

**Solution:**

To find the volume \( V \), we shall follow these steps:

1. **Identify the region of integration:** Determine the points of intersection of the curves.
2. **Set up the integral using the method of cylindrical shells:** The formula for the volume using cylindrical shells is:
\[
V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx
\]

Here, \( f(x) \) is the height of the shell and \( x \) is the radius of the shell.

3. **Evaluate the integral:** Compute the integral to find the volume.

---

**Please Solve:**

\[ V = \boxed{\phantom{\text{Solution Here}}} \]

By completing the calculation, we will obtain the volume of the solid.
Transcribed Image Text:### Using the Method of Cylindrical Shells **Problem:** Use the method of cylindrical shells to find the volume \( V \) generated by rotating the region bounded by the given curves about the \( y \)-axis. \[ y = 2 + x - x^2, \quad x + y = 2 \] **Solution:** To find the volume \( V \), we shall follow these steps: 1. **Identify the region of integration:** Determine the points of intersection of the curves. 2. **Set up the integral using the method of cylindrical shells:** The formula for the volume using cylindrical shells is: \[ V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx \] Here, \( f(x) \) is the height of the shell and \( x \) is the radius of the shell. 3. **Evaluate the integral:** Compute the integral to find the volume. --- **Please Solve:** \[ V = \boxed{\phantom{\text{Solution Here}}} \] By completing the calculation, we will obtain the volume of the solid.
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