Find the volume V of the described solid S.
The base of S is a circular disk with radius 2r. Parallel cross-sections perpendicular to the base are squares.
To find the volume of the solid, we need to integrate the areas of the square cross-sections over the height of the solid.
Let's first consider a single square cross-section at a height h from the base. The side length of this square cross-section is equal to the diameter of the circular disk, which is 2(2r) = 4r.
The area of a square is given by the formula A = side length^2. Therefore, the area of this square cross-section is A = (4r)^2 = 16r^2.
To find the volume V, we need to integrate the areas of all the square cross-sections over the height of the solid S. The height of the solid is not given, so let's assume it is h_max.
Therefore, the integral for the volume V is:
V = ∫[0 to h_max] 16r^2 dh
Integrating this expression gives:
V = 16r^2 * ∫[0 to h_max] dh
V = 16r^2 * [h] evaluated from 0 to h_max
V = 16r^2 * (h_max - 0)
V = 16r^2 * h_max
So, the volume V of the described solid S is 16r^2 * h_max.
To find the volume V of the described solid S, follow these steps:
Step 1: Visualize the solid
The solid S is described as having a circular disk as its base with a radius of 2r. Additionally, the parallel cross-sections perpendicular to the base are squares. So, imagine a stack of squares standing on the circular disk base.
Step 2: Understand the relationship between the circular base and the squares
Since the squares are parallel cross-sections and perpendicular to the base, each square will have a side length equal to the diameter of the circular base. The diameter of the circular base is twice the radius, so each square has a side length of 2r.
Step 3: Visualize the slicing of the solid
To calculate the volume, imagine slicing the solid S into infinitely thin parallel slices perpendicular to the base, starting from the base and moving up to the top. Each slice will be a square with side length 2r.
Step 4: Calculate the volume of each slice
Since each slice is a square with side length 2r, the area of each slice is (2r)^2 = 4r^2.
Step 5: Sum up the volumes of all the slices
To find the volume of the entire solid S, you need to sum up the volumes of all the slices. Since the slices are infinite and infinitely thin, this effectively means integrating the volume of each slice over the region it covers.
Step 6: Set up the integral
Since the slices are parallel, the integration can be done with respect to the variable z, where z represents the height of each slice from the base. The range of z will be from 0 (the height of the base) to h (the height of the solid), so the integral will be ∫[0,h] 4r^2 dz.
Step 7: Evaluate the integral
Evaluating the integral ∫[0,h] 4r^2 dz will give you the volume of the solid S. The integral of 4r^2 with respect to z is just 4r^2z. Evaluating this integral over the range [0,h] gives the volume V = 4r^2h.
Therefore, the volume V of the described solid S is 4r^2h.
I may have this wrong, but here's how I see it:
Draw a circle of radius 2r. That's the base. Draw a chord perpendicular to the x-axis at distance x from the center. At this point,
x^2 + y^2 = 4r^2
Now, erect a square of height 2y on the base. That is a cross-section of the solid. It has area 4y^2
The volume is thus Int(4y^2 dx)[0,2r]
But, y^2 = 4r^2 - x^2
v = 4*Int(4r^2 - x^2) dx)[0,2r]
= 4*(4r^2 x - 1/3 x^3)[0,2r]
= 4(8r^3 - 8r^3/3)
= 32(2/3 r^3)
= 64/3 r^3