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!