Use the general slicing method to find the volume of the following solid. 
The solid whose base is the region bounded by the curve y equals 20 StartRoot cosine x EndRoot and the​ x-axis on left bracket negative StartFraction pi Over 2 EndFraction comma StartFraction pi Over 2 EndFraction right bracket
​, and whose cross sections through the solid perpendicular to the​ x-axis are isosceles right triangles with a horizontal leg in the​ xy-plane and a vertical leg above the​ x-axis.
y equals 20 StartRoot cosine x EndRoot
x
y
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Part 1
Set up the integral that gives the volume of the solid. Use increasing limits of integration. Select the correct choice below and fill in the answer boxes to complete your choice.
​(Type exact​ answers.)

To find the volume of the solid using the general slicing method, we'll need to follow these steps:

### 1. **Understand the Geometry of the Cross Section**
- The base of the solid is the region bounded by the curve \( y = 20\sqrt{\cos(x)} \) and the x-axis, on the interval \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \).
- The cross sections perpendicular to the x-axis are isosceles right triangles with a horizontal leg in the xy-plane and a vertical leg above the x-axis.

### 2. **Determine the Area of a Cross Section**
- For an isosceles right triangle with a horizontal leg in the xy-plane:
  - The length of the horizontal leg (which lies on the xy-plane) is \( 2y \), since the horizontal leg extends from the x-axis to \( y = 20\sqrt{\cos(x)} \).
  - The vertical leg, by symmetry, also has length \( 2y \).
  - The area \( A(x) \) of such a triangle is given by:
    \[
    A(x) = \frac{1}{2} \times (\text{leg}) \times (\text{leg}) = \frac{1}{2} \times 2y \times 2y = 2y^2
    \]

### 3. **Express the Area in Terms of x**
- Substitute \( y = 20\sqrt{\cos(x)} \) into the expression for \( A(x) \):
  \[
  A(x) = 2 \left( 20\sqrt{\cos(x)} \right)^2 = 2 \times 400 \cos(x) = 800 \cos(x)
  \]

### 4. **Set Up the Integral for the Volume**
- The volume \( V \) of the solid is obtained by integrating the area \( A(x) \) from \( x = -\frac{\pi}{2} \) to \( x = \frac{\pi}{2} \):
  \[
  V = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} A(x) \, dx = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 800 \cos(x) \, dx
  \]

### 5. **Compute the Integral**
- We now compute the integral:
  \[
  V = 800 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos(x) \, dx
  \]
- The integral of \( \cos(x) \) is \( \sin(x) \), so:
  \[
  V = 800 \left[ \sin(x) \right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}}
  \]
- Evaluate the integral at the limits:
  \[
  V = 800 \left( \sin\left(\frac{\pi}{2}\right) - \sin\left(-\frac{\pi}{2}\right) \right) = 800 \left( 1 - (-1) \right) = 800 \times 2 = 1600
  \]

### Final Answer:
The volume of the solid is \( 1600 \) cubic units.

Do you want details or have any questions?

Here are 5 related questions:
1. How would the volume change if the cross sections were squares instead of isosceles right triangles?
2. What if the base curve was \( y = \sin(x) \) instead of \( y = 20\sqrt{\cos(x)} \)?
3. Can you explain why the limits of integration were chosen as \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \)?
4. How would the process differ if the solid was rotated around the x-axis?
5. What if the cross sections were equilateral triangles instead?

**Tip:** Always make sure the area function \( A(x) \) correctly represents the cross-sectional area, which depends on the geometry given in the problem.

Use the general slicing method to find the volume of the following solid. The solid whose base is the region bounded by the curve y equals 20 StartRoot cosine x EndRoot and the​ x-axis on left bracket negative StartFraction pi Over 2 EndFraction comma StartFraction pi Over 2 EndFraction right bracket​, and whose cross sections through the solid perpendicular to the​ x-axis are isosceles right triangles with a horizontal leg in the​ xy-plane and a vertical leg above the​ x-axis.

Learn how to calculate the volume of a solid with a base bounded by the curve y = 20√cos(x) and cross sections that are isosceles right triangles. This problem involves the general slicing method and is suitable for students in Grades 11-12 or early college.

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