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To compute the volume of the solid whose base is the region in the first quadrant bounded by \( y = x^6 \), \( y = 1 \), and the \( y \)-axis, and whose cross-sections perpendicular to the \( y \)-axis are equilateral triangles, we can break the problem down step by step.

### Step-by-Step Solution:

1. **Cross-sectional Area:**
   The cross-sections are equilateral triangles. The side length of each equilateral triangle depends on the value of \( x \) for each \( y \)-value.

   Since the equation \( y = x^6 \) defines the relationship between \( x \) and \( y \), we can solve for \( x \) as a function of \( y \):
   \[
   x = y^{1/6}
   \]
   This gives us the base of the equilateral triangle for each value of \( y \).

2. **Area of an Equilateral Triangle:**
   The area \( A \) of an equilateral triangle with side length \( s \) is given by:
   \[
   A = \frac{\sqrt{3}}{4} s^2
   \]
   Here, the side length of the triangle is \( 2x = 2y^{1/6} \). So, the area becomes:
   \[
   A(y) = \frac{\sqrt{3}}{4} (2y^{1/6})^2 = \frac{\sqrt{3}}{4} \cdot 4y^{1/3} = \sqrt{3} y^{1/3}
   \]

3. **Volume Integral:**
   The volume \( V \) of the solid is obtained by integrating the cross-sectional area \( A(y) \) from \( y = 0 \) to \( y = 1 \):
   \[
   V = \int_0^1 A(y) \, dy = \int_0^1 \sqrt{3} y^{1/3} \, dy
   \]

4. **Evaluate the Integral:**
   We can now compute the integral:
   \[
   V = \sqrt{3} \int_0^1 y^{1/3} \, dy
   \]
   The antiderivative of \( y^{1/3} \) is:
   \[
   \frac{3}{4} y^{4/3}
   \]
   So the integral becomes:
   \[
   V = \sqrt{3} \left[ \frac{3}{4} y^{4/3} \right]_0^1 = \sqrt{3} \cdot \frac{3}{4} \cdot \left( 1^{4/3} - 0^{4/3} \right)
   \]
   \[
   V = \sqrt{3} \cdot \frac{3}{4} = \frac{3\sqrt{3}}{4}
   \]

### Final Answer:
\[
V = \frac{3\sqrt{3}}{4}
\]

Let me know if you need any clarifications or further details!

### Related Questions:
1. How do you calculate the volume of solids with different cross-sectional shapes?
2. What happens if the cross-sectional shape is not equilateral but isosceles?
3. Can you compute the volume of the solid if the base is different, say a semicircle?
4. How does changing the power in the equation \( y = x^n \) affect the volume?
5. How do you approach volumes when the cross-section is a function of two variables?

### Tip:
For volumes of solids of known cross-sections, always identify the relationship between the base dimensions and the cross-sectional area formula. This makes setting up the integral easier!

Compute the volume of the solid whose base is the region in the first quadrant bounded by y = x^6, y = 1, and the y-axis, and whose cross sections perpendicular to the y-axis are equilateral triangles.

Learn how to compute the volume of a solid with base bounded by y = x^6 and y = 1 in the first quadrant, with cross-sections perpendicular to the y-axis being equilateral triangles. This calculus problem involves integrating the area of equilateral triangles to find the volume.

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