Questions: Use the washer method to find the volume of the solid generated by revolving the shaded region about the x-axis. V= (Type an exact answer, using π as needed.)

Use the washer method to find the volume of the solid generated by revolving the shaded region about the x-axis.

V=

(Type an exact answer, using π as needed.)

Transcript text: Use the washer method to find the volume of the solid generated by revolving the shaded region about the $x$-axis. \[ V= \] (Type an exact answer, using $\pi$ as needed.)

Solution

Solution Steps

To solve the problem of finding the volume of the solid generated by revolving the shaded region about the $ x $-axis using the washer method, we will follow these steps:

Step 1: Identify the Functions and Limits

First, identify the functions that bound the shaded region and the limits of integration. Let's assume the region is bounded by the functions $ y = f(x) $ and $ y = g(x) $, where $ f(x) \geq g(x) $ over the interval $[a, b]$.

Step 2: Set Up the Washer Method Formula

The volume $ V $ of the solid of revolution using the washer method is given by the integral:

\[ V = \pi \int_a^b \left( [f(x)]^2 - [g(x)]^2 \right) \, dx \]

Step 3: Evaluate the Integral

Substitute the functions $ f(x) $ and $ g(x) $ into the formula and evaluate the integral over the interval $[a, b]$.

Final Answer

The exact volume of the solid is:

\[ \boxed{V = \pi \int_a^b \left( [f(x)]^2 - [g(x)]^2 \right) \, dx} \]

This is the general form of the solution. To provide a specific answer, the functions $ f(x) $, $ g(x) $, and the limits $ a $ and $ b $ need to be specified.

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