Answer:
Explanation:
To find the volume of the solid, we need to integrate the areas of the cross-sections perpendicular to the x-axis. In this case, the cross-sections are semicircles.
Let's set up the integral to find the volume. The base of the solid is the region in the first quadrant bounded by y = x^6, y = 1, and the y-axis. We will integrate along the x-axis from x = 0 to x = 1.
The area of each semicircle is given by (1/2) * π * r^2, where r is the radius of the semicircle.
Now, let's find the radius of each semicircle as a function of x. Since the base is bounded by y = x^6 and y = 1, the radius r is simply the y-coordinate at each x, which is r = x^6.
The volume of the solid is then given by the integral of the areas of the semicircles:
Volume = ∫[0 to 1] (1/2) * π * (x^6)^2 dx
Volume = ∫[0 to 1] (1/2) * π * x^12 dx
Now, integrate with respect to x:
Volume = (1/2) * π * ∫[0 to 1] x^12 dx
Volume = (1/2) * π * [x^13 / 13] evaluated from 0 to 1
Volume = (1/2) * π * [(1^13 / 13) - (0^13 / 13)]
Volume = (1/2) * π * (1/13)
Volume ≈ 0.03832 cubic units (rounded to five decimal places)
So, the volume of the solid is approximately 0.03832 cubic units.