Find the volume V of the described solid S.
The base of S is a circular disk with radius 4r. Parallel cross-sections perpendicular to the base are squares.
using symmetry,
v = 4∫[0,4r]∫[0,√(16r^2-x^2)] 4xy dy dx
= 512r^4
To find the volume V of the described solid S, we need to break it down into smaller, more manageable sections.
First, let's consider the base of S, which is a circular disk with radius 4r. The formula for the area of a circle is A = πr^2, where r is the radius. Therefore, the area of the base of S is A = π(4r)^2.
Next, we need to consider the parallel cross-sections perpendicular to the base. These cross-sections are squares. Since the base is a circular disk, we can imagine slicing S into thin slices parallel to the base. Each of these slices will have the same area as the base.
Let's consider one of these slices. Since the cross-section is a square, the area of this slice is the square of the length of one side. The question states that the cross-sections are squares, but it doesn't provide any information about the side length. Let's call the side length s.
To find the side length s, we can consider the radius 4r of the circular base. The diameter of the circular base is twice the radius, so it is 2(4r) = 8r. Note that the side length of the square cross-section will be equal to the diameter of the circular base.
The formula for the diagonal of a square is d = s√2, where d is the diagonal length and s is the side length. In this case, the diagonal length is equal to the diameter of the circular base, which is 8r. Therefore, we can set up the equation 8r = s√2 and solve for s:
8r = s√2
s = (8r) / √2
s = 8r / √2
Now that we have the side length s of the square cross-section, we can find its area. The formula for the area of a square is A = s^2, where s is the side length. Therefore, the area of the square cross-section is A = (8r / √2)^2.
To find the volume V of the solid S, we need to integrate the areas of all the slices. Since the slices are parallel, we can think of them as infinitesimally thin disks stacked on top of each other. Therefore, we can integrate the areas using the formula:
V = ∫A dx
where A is the area of each slice, and dx is the variable representing the thickness of the infinitesimal slice.
To set up the integral, we need to express the area A in terms of x, the variable along the height of the disk. But since the cross-sections are squares, which have equal lengths on each side, the area of each slice does not depend on x. Therefore, A is just a constant.
Let's call the thickness of the infinitesimal slice dx. The height h of the solid S is the same as the radius of the base, so we can set up the integral as follows:
V = ∫A dx
V = ∫(π(4r)^2 dx)
V = ∫(16πr^2 dx)
V = 16πr^2 ∫dx
V = 16πr^2x + C
where C is the constant of integration.
Therefore, the volume V of the solid S is given by the equation V = 16πr^2x + C.