Do I have a legitimate shortcut for finding the area under a curve?
I'm studying functions, and recently I've spent some time looking into finding the area under a parabolic curve with integral calculus.
The methods I've found in this regard are very straightforward, though (and this must be because of a limited knowledge of more advanced methods) I find the solution can take quite some time to write up.
I've toyed around with what seems to be a very simple setup to get an accurate solution to the area under a curve, as well as the area between a curve and a right triangle...
Parabola represents 1/4๐ฅ^2
๐ฅ1 = 0 units
๐ฆ1 = 0 units
๐ฅ2 = 5 units
๐ฆ2 = 1/4(5)^2 = 6.25 units
Regarding one-half of the figure above:
Area of the rectangle, R; from ๐ฅ = 0 to ๐ฅ = 5; to ๐ฆ = 6.25:
๐ด = ๐ โ ๐ค
๐ด = 6.25 โ 5
๐ด = 31.25 units
Area of the triangle; A:
๐ด = 1/2 โ ๐ โโ
๐ด = 1/2 โ 5 โ 6.25
๐ด = 15.625 units
Area of under-curve; B:
๐ด๐ = ๐ด1 + ๐ด2
๐ด๐ = 1/6 โ 2(๐ฅ2 โ ๐ฅ1) โ (๐ฆ2 โ ๐ฆ1) + (๐ฆ1 โ ๐ฅ2)
๐ด๐ = 1/6 โ 2(5 โ 0) โ (6.25 โ 0) + (0 โ 5)
๐ด๐ = 1/6 โ (10) โ (6.25) + (0)
๐ด๐ = 10.41666667 units
Derived from Archimedesโ assertion that the area of the inside of a parabola is equal to 2/3๐โ. ๐ด2 describes the addition of the area of an inner rectangle that would result from a vertical phase shift; for example, a positive c-value in the form ๐๐ฅ^2+๐๐ฅ+๐. In this case, ๐ฆ1 begins at a number >0, which creates a rectangle directly under the bottom edge of section B.
Area of the space between the triangle and curve; C:
๐ด๐ถ = ๐
โ (๐ด + ๐ต)
๐ด๐ถ = 31.25 โ (15.625 + 10.41666667)
๐ด๐ถ = 5.20833333 units
2/3๐โ is 2/3 of the area of the triangle, and 1/3 of the area of the rectangle.
The area of C is proportionate to 5.20833333/15.625 = 1/3 of the area of A.
The area of C is proportionate to 5.20833333/10.41666667 = 1/2 of the area of B.
Equation for determining the area of C:
๐ด = 1/6๐โ
๐ด = 1/6(5)(6.25)
๐ด = 5.208333333 units
I'm very sorry if the formatting doesn't work out for this post. I'm not used to the layout, nor am I (as you can tell) a very effective communicator of math (haha). Unless there's a '^' involved, any number following a variable is meant as subscript. I will re-post more carefully or email a .pdf to anybody who may have any problems wading through this. Thank you for any advice and tips!
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