let a hemisphere of radium 5 be cut by a plane parallel to the base of the hemisphere thus forming a segment of height 2. find its volume using cylindrical shells... thanks
To find the volume of this segment using cylindrical shells, we need to integrate the volume of each cylindrical shell.
First, let's visualize the shape of the segment. It will be a cylindrical segment with a height of 2 and a curved surface area forming the top of the segment.
The radius of the segment's base will be the same as the radius of the hemisphere, which is 5 units.
To integrate the volume, we'll consider a small cylindrical shell with a height of Δh and a radius of r. The volume of this cylindrical shell can be approximated as πr^2Δh.
Since the radius of the segment's base changes as we move up, we need to express the radius r as a function of height h.
The height of the segment ranges from 0 to 2, and at each height, h, the radius of the segment can be calculated using the equation:
r = √(5^2 - h^2)
To find the total volume of the segment, we integrate the volume formula over the height range:
V = ∫[0 to 2] πr^2 dh
Substituting the expression for r, we have:
V = ∫[0 to 2] π(5^2 - h^2) dh
Simplifying this integral, we get:
V = ∫[0 to 2] π(25 - h^2) dh
Now, we can evaluate this integral:
V = π * ∫[0 to 2] (25h - h^3) dh
V = π * (25 * ∫[0 to 2] h dh - ∫[0 to 2] h^3 dh)
V = π * (25 * [h^2/2] [0 to 2] - [h^4/4] [0 to 2])
V = π * (25 * (2^2/2) - (2^4/4))
V = π * (25 * 2 - 4)
V = π * (50 - 4)
V = π * 46
Therefore, the volume of the segment is 46π cubic units.
To find the volume of the segment formed by cutting a hemisphere with a plane parallel to its base, we can calculate it using cylindrical shells. Here's how to do it:
1. Visualize the Hemisphere: Start by visualizing a hemisphere with a radius of 5 units.
2. Find the Height of the Segment: The height of the segment is given as 2 units. This means the cut-off part of the hemisphere starts at the base and extends 2 units above.
3. Set up the Integral: To calculate the volume using cylindrical shells, we need to integrate the cross-sectional area of each shell from the base to the top of the segment. Since we are working with cylindrical shells, we'll integrate with respect to the height.
4. Choose a Variable: Let's choose 'h' as our variable for the height of the segment. We'll integrate from 0 to 2 since the height of the segment is 2 units.
5. Determine the Radius of Each Shell: As we move up from the base to the top of the segment, the radius of each cylindrical shell decreases linearly. To find the radius 'r' of each shell at height 'h', we can use similar triangles.
At height 'h', the radius 'r' can be found using the formula:
r = (5/2) - (h/2)
6. Calculate the Cross-sectional Area: The cross-sectional area of each shell is given by the formula for the area of a circle, A = πr^2.
Substituting the expression for 'r' from step 5:
A = π[(5/2) - (h/2)]^2
7. Integrate: Finally, integrate the cross-sectional area with respect to 'h' from 0 to 2:
V = ∫[0 to 2] π[(5/2) - (h/2)]^2 dh
8. Evaluate the Integral: Evaluate the integral to find the volume of the segment. Simplify the expression and integrate term by term:
V = π∫[0 to 2] [(25/4) - 5h + (h^2/4)] dh
V = π[(25h/4) - (5h^2/2) + (h^3/12)] [0 to 2]
V = π[(25(2)/4) - (5(2)^2/2) + (2^3/12)] - π[(25(0)/4) - (5(0)^2/2) + (0^3/12)]
V = π(50/4) - π(20/2) + π(8/12)
V = (25/2)π - 10π + (2/3)π
V = (25π - 60π + 8π)/6
V = (-17π)/6
Therefore, the volume of the segment cut from the hemisphere is (-17π)/6 cubic units.