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 1/4π₯^2
π₯1 = 0 units
π¦1 = 0 units
π₯2 = 5 units
π¦2 = 1/4(5)^2 = 6.25 units
Regarding one-half of the figure above:
Area of the rectangle, R; from π₯ = 0 to π₯ = 5; to π¦ = 6.25:
π΄ = π β π€
π΄ = 6.25 β 5
π΄ = 31.25 units
Area of the triangle; A:
π΄ = 1/2 β π ββ
π΄ = 1/2 β 5 β 6.25
π΄ = 15.625 units
Area of under-curve; B:
π΄π = π΄1 + π΄2
π΄π = 1/6 β 2(π₯2 β π₯1) β (π¦2 β π¦1) + (π¦1 β π₯2)
π΄π = 1/6 β 2(5 β 0) β (6.25 β 0) + (0 β 5)
π΄π = 1/6 β (10) β (6.25) + (0)
π΄π = 10.41666667 units
Derived from Archimedesβ assertion that the area of the inside of a parabola is equal to 2/3πβ. π΄2 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 ππ₯^2+ππ₯+π. In this case, π¦1 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:
π΄πΆ = π
β (π΄ + π΅)
π΄πΆ = 31.25 β (15.625 + 10.41666667)
π΄πΆ = 5.20833333 units
2/3πβ 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:
π΄ = 1/6πβ
π΄ = 1/6(5)(6.25)
π΄ = 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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