AIM OF THE EXPLORATION
ÿ Explore Integral Calculus usage to find the Volume of the solids
ÿ Identify the cross-section of the solid
ÿ Suggest an Algorithm for finding an expression for the Volume
INTRODUCTION
We learnt in the Applications of the Integral calculus to find the area under the curve.
This can be divided in following three cases:
ÿ Area below any given curve and above the X-axis
ÿ Area between the two given curves
If definite integration can be used to calculate the area of any figure in XY plane, then there must be
some way to calculate the volume of Figures in 3 Dimensional Geometry. can calculus be used for this
purpose.
Yes definitely, and that is the topic of our exploration. We will try to demonstrate the
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Below is a sketch of
the problem.
This problem is relatively straightforward to set up. Using our “top curve minus bottom curve”
idea we obtain the integral
Volumes with known cross sections
The method of finding the volume of a solid with a known cross section is, very straightforward. We will
find an expression for the volume of a slice of the solid and then integrate this expression over a specified
interval. We have learnt that the definite integral allows us to sum an infinite number of objects. It is like
finding the volume of a bread by finding the volume of one very thin slice and then adding up an infinite
number of these slices—all of which is packed between the two ends of the bread.
CASE I : SOLID WITH CROSS-SECTION OF A SQUARE
Consider the region bounded by f (x) = √x, the x-axis and x = 4. This region will be the base of a solid
whose cross section, perpendicular to the x-axis, is a square. We will imagine the x-y coordinate plane
lying flat. If we slice this solid and look from one end of it, then we will see a square. Fig below depicts
the view of the object with some "slices" perpendicular to the x-y plane.
To visualize this diagram completely and easily, we have to imagine the standard coordinate
plane with the y-axis "pulled" toward
teacher I am required to take into account both dimensions. So this became my goal and tool to use.
Purpose: Weighing objects. Figuring out the density with an object by calculated volume and Archimedes’ Principle.
Can I apply this somehow to volume? Well at standard temperature and pressure (STP) a mole of a gas will occupy 22.4 liters. So If we keep our units straight we should be able calculate a given volume of gas from moles. Check it out…… http://www.sciencegeek.net/Chemistry/Video/Unit4/GMV4.shtml
14. Calculate the volume of the magnet by multiplying the length × width × height, record in Data Table 5.
The perimeter of a square is 8 inches. What is the area of the square if each side is a whole number?
heavy duty samples as well as your dimensional measurements (length and width in cm) from Part III of this experiment, calculate the height, or thickness, of each sample of aluminum using the formula V l x w x h. In the formula, V stands for volume, l for length, w for width, and h for height. Once again, you will have to use your algebraic skills to manipulate the formula, to solve for height. You must show all your work. (15 pts)
reduce the sides and then use my first equation and It worked! So I tried it
Step 1: The volume of the quarter-sphered tank rounds up to 478,676. You use the condition to find the volume.
We used this equation to generate different heights that would maximize the area of the cuboid. We were able to isolate just the area of the cuboid because that one face is simply projected backwards 150m, so we mathematically we were able to ignore that constant in the volume equation shown above. With the help of Excel spreadsheets, we calculated the
To achieve a good volumetric technique, the experimenter needs to be able to correctly complete certain procedures.
This purpose of this experiment is to calculate the thickness of a sheet of aluminum foil. This experiment is necessary because the human eye cannot accurately measure the small thickness of aluminum foil with only a ruler. However, to understand the procedure one needs to understand conversion, density, and volume. Conversion is when one converts one unit to another unit using a conversion factor(e.g. 2.54cm/in). Density is how much mass there is in a certain volume(density=mass/volume) and it stays constant in a substances and mixtures that have the same composition. Volume is the amount of space that an object occupies. The experiment will consist of weighing of aluminum foil, measuring the length and width, then converting these values
In order to obtain the true volume of volumetric glassware holds, this formula will be used.
A square comes out to be a two-dimensional object, but it doesn’t work out so cleanly with the Sierpinski Triangle. When you divide the triangle into pieces whose sides are half the size of the original, you get three self-similar triangles, so the formula works out:
⇒ The investigation of the area begins by supposing that C0, the initial curve (an equilateral triangle) has a total area of 1 unit2.