Method of Quadrature and distance
Finding areas of a circle by estimating the amount of area under a curve by "blocking" the area on a coordinate plane is called finding areas with the "method of quadrature."
This method gives 1/4 of the area of a circle. The right hand rule over-estimates the area because the rectangular areas are outside the section of the circle being used to estimate the area. The left-hand rule under-estimates the area because the rectangular areas are inside the section of the area of the circle being used to estimate the area. With the left-hand rule, part of the area is not calculated.
By using a combination of both rules, one can find a balance between the unused portions and the excess portions. This gives a closer estimate.
The distance formula is derived from the Pythagorean theorem.
Let "a" be the length of one side of a triangle, "b" be the length of the second side of a triangle, and "c" be the length of the hypothenuse.
If we are given two points on a number line, the length of "a" is the difference between the two points.
Example: A line on the x axis that lies between points -3 and 4 has a length of the absolute value of -3 - 4 or 4 - (-3) which equals 7.
If we are given two points on a coordinate plane, we must consider both coordinates (x and y).
Example: (0, 3) and (4, 0)
Find the difference between the x coordinates, 0 and 4. 0 - 4 = -4. Length cannot be negative, so we must use the absolute value which is 4.
Now find the difference between the y coordinates 3 and 0. 3 - 0 = 3.
We now have the length of two sides of a triangle, but we do not have the distance between the two points. To determine the distance between the two points we must find the hypotenuse formed by connecting the 'end points' of the legs. Ergo, we use the pythagorean theorem.
The distance is found by taking the square root of the sum of the lengths of the sides.