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 1/4^2
1 = 0 units
1 = 0 units
2 = 5 units
2 = 1/4(5)^2 = 6.25 units
Regarding one-half of the figure above:
Area of the rectangle, R; from = 0 to = 5; to = 6.25:
= ∗
= 6.25 ∗ 5
= 31.25 units
Area of the triangle; A:
= 1/2 ∗ ∗ℎ
= 1/2 ∗ 5 ∗ 6.25
= 15.625 units
Area of under-curve; B:
= 1 + 2
= 1/6 ∗ 2(2 − 1) ∗ (2 − 1) + (1 ∗ 2)
= 1/6 ∗ 2(5 − 0) ∗ (6.25 − 0) + (0 ∗ 5)
= 1/6 ∗ (10) ∗ (6.25) + (0)
= 10.41666667 units
Derived from Archimedes’ assertion that the area of the inside of a parabola is equal to 2/3ℎ. 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 ^2++. In this case, 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:
= − ( + )
= 31.25 − (15.625 + 10.41666667)
= 5.20833333 units
2/3ℎ 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:
= 1/6ℎ
= 1/6(5)(6.25)
= 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!