AIM OF THE EXPLORATION ÿ Explore Integral Calculus usage to find the Volume of the solids ÿ

1000 WordsApr 23, 20194 Pages
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…show more content…
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

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