Find the volume of the solid generated by revolving the region bounded by y equals 3 StartRoot Bold sin x EndRoot​,  y equals 0 comma and x 1 equals StartFraction pi Over 4 EndFraction
 and x 2 equals StartFraction 3 pi Over 4 EndFraction
 about the​ x-axis.
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Part 1
The volume of the solid is 
  
enter your response here cubic units.
​(Type an exact​ answer, using pi as​ needed.)

To find the volume of the solid generated by revolving the region bounded by \( y = 3\sqrt{\sin(x)} \), \( y = 0 \), \( x_1 = \frac{\pi}{4} \), and \( x_2 = \frac{3\pi}{4} \) around the x-axis, we can apply the **disk method**. 

### Disk Method Formula:
\[
V = \pi \int_{x_1}^{x_2} [f(x)]^2 \, dx
\]
Where \( f(x) = y = 3\sqrt{\sin(x)} \), and we are revolving around the x-axis from \( x_1 = \frac{\pi}{4} \) to \( x_2 = \frac{3\pi}{4} \).

### Step 1: Set up the integral
The radius of each disk is given by \( y = 3\sqrt{\sin(x)} \), so:
\[
V = \pi \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \left(3\sqrt{\sin(x)}\right)^2 \, dx
\]

### Step 2: Simplify the integrand
\[
\left(3\sqrt{\sin(x)}\right)^2 = 9\sin(x)
\]
Thus, the integral becomes:
\[
V = \pi \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} 9\sin(x) \, dx
\]

### Step 3: Compute the integral
\[
V = 9\pi \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \sin(x) \, dx
\]
The antiderivative of \( \sin(x) \) is \( -\cos(x) \), so:
\[
V = 9\pi \left[-\cos(x)\right]_{\frac{\pi}{4}}^{\frac{3\pi}{4}}
\]

### Step 4: Evaluate the definite integral
First, calculate \( \cos(x) \) at the bounds:
\[
\cos\left(\frac{3\pi}{4}\right) = -\frac{1}{\sqrt{2}}, \quad \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}
\]

Now, plug these values into the expression:
\[
V = 9\pi \left[-\left(-\frac{1}{\sqrt{2}}\right) - \left(-\frac{1}{\sqrt{2}}\right)\right]
\]
\[
V = 9\pi \left[\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}\right]
\]
\[
V = 9\pi \left[\frac{2}{\sqrt{2}}\right]
\]
\[
V = 9\pi \times \sqrt{2}
\]

### Final Answer:
The volume of the solid is \( 9\pi\sqrt{2} \) cubic units.

Would you like more details on this solution, or do you have any other questions?

Here are five related questions to deepen your understanding:

1. How does the disk method differ from the washer method in finding volumes?
2. What would change in the integral if the region were revolved around the y-axis instead?
3. How does the behavior of \( \sin(x) \) affect the shape of the solid of revolution?
4. Can you apply the cylindrical shell method to this problem? If so, how?
5. What is the effect of changing the limits of integration on the volume?

**Tip**: When setting up integrals for solids of revolution, always check whether the function you're revolving requires using the disk or washer method!

Find the volume of the solid generated by revolving the region bounded by y = 3√sin(x), y = 0, x₁ = π/4, and x₂ = 3π/4 about the x-axis.

Learn how to calculate the volume of the solid formed by revolving the curve y = 3√sin(x) around the x-axis using the disk method. This detailed solution explains each step, from setting up the integral to evaluating it, and includes related concepts like trigonometric integration and the disk method formula. Suitable for Grades 11-12 or college-level calculus students.

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