source: http://indusladies.com/community/threads/dot-kolams.77039/page-2 Can we use integrals to figure out the area of the insides of such complicated designs and shapes? Well, to introduce you to these designs, called 'kolams', a typical design drawn during the season of lights and happiness, I have posted a picture above for you to get the idea. So, lets say we had a simple kolam to deal with at first, such as this one: source: http://www.a1tamilnadu.com/singlePost.asp?cid=1199#.Wgj22bA-dPM Lets say the problem was to find the area inside of this complex shape. I want to try to use integrals to find the area of this shape. So, therefore, lets use a small, tiny arbitrary curve from this whole design. In other words, take a magnifying glass and find a curvature in which we can find a small approximation of the area inside that small curve. So, lets pick a small curve from this kolam and draw a x-y axis behind it to integrate it and approximate the area. This curve looks a lot like a natural log curve. We can use the left rectangular approximation method to find the approximate area. if we integrate ln(x) from x = 1 to x = 4, we get 2.5451.... ∫ [at interval 1,4] ln(x) d(x) = 2.5451.. However, lets try using rectangles to approximate the areas. So, for the first rectangle (from x=1 to x=2), we can solve the area, A, by getting the y value at x = 1 and multiplying it by one to get the area of that particular rectangle. Therefore, the area of the first rectangle would be ln(1) which is the y value, times 1, which yields 0. The next rectangle (from x=2 to x=3) is ln(2) times again 1 (which represents the width) which yields 0.6. The next rectangle (from x=3 to 4) is ln(3) times 1 = 1.09. Therefore, the total area which encompasses the area under the curve is: 1.09 + 0.6 + 0 = 1.7917, much lower than the actual area calculated above (2.5451). So, how do we solve for the area inside of these designs? It may seem a tedious task, but to be exact and accurate we must calculate the areas underneath the smaller curves of the design to eventually find the total area the design covers. In order to do this, we will have to put the design on to a grid and label the coordinates.
Good luck!!
0 Comments
|
Archives
May 2021
Topics
All
|