Find the volume of the solid obtained by rotating the region bounded by y=x^3, y=1, and the y-axis and whose cross-sections perpendicular to the y axis are equilateral triangles.
I asked this same question for the y-axis around the x-axis (Thanks for the explanation) but I don't get how to solve this one either.
I get 2(sqrt3/2)(1-(3/4)1^(4/3))
But that's not the answer.
To find the volume of the solid obtained by rotating the region bounded by y = x^3, y = 1, and the y-axis, with cross-sections perpendicular to the y-axis as equilateral triangles, we will need to integrate.
First, let's draw the given region in the x-y plane. This region is bounded by the cubic curve y = x^3, the horizontal line y = 1, and the y-axis.
Since we have cross-sections perpendicular to the y-axis, we will integrate with respect to y. We need to determine the limits of integration for y. Looking at the graph, we can see that the region lies between y = 0 and y = 1.
Next, let's consider a small element of thickness dy at a distance y from the y-axis. Each cross-section at this distance is an equilateral triangle. We need to determine the side length of this equilateral triangle.
For an equilateral triangle, all sides are equal. Let's denote the side length of the equilateral triangle by s. Since the cross-section is perpendicular to the y-axis, the base of the triangle will be equal to s, and the height will be equal to 2s/√3 (as the equilateral triangle is divided into two 30-60-90 right triangles).
Now, let's find the relation between s and y. To do this, we need to consider the equation of the curve y = x^3. We know that the base of the equilateral triangle is equal to s. Therefore, the corresponding value of x can be found by taking the cube root of y, which gives us x = ∛y.
Since the height of the triangle is 2s/√3, and the distance from the y-axis to the curve y = x^3 is x = ∛y, we can set up the equation:
2s/√3 = ∛y
Solving for s, we get:
s = √3(∛y)/2
Now that we have the side length of the equilateral triangle, we can find the area of the cross-section at distance y, by using the formula for the area of an equilateral triangle:
Area = (sqrt(3) / 4) * s^2
Substituting the expression for s, we obtain:
Area = (sqrt(3) / 4) * (√3(∛y)/2)^2
Area = (3 / 4) * (√3y^(2/3))
Finally, to find the volume, we integrate the area function over the interval y = 0 to y = 1:
Volume = ∫[0,1] (3 / 4) * (√3y^(2/3)) dy
Evaluating this integral will give us the volume of the solid obtained by rotating the region bounded by y = x^3, y = 1, and the y-axis, with cross-sections perpendicular to the y-axis being equilateral triangles.