5.6 TheDistance Formula and the Method of Quadrature

Geometry

Holt, Rinehart, and Winston

Objectives :

* develop and apply the distance formula

* Use the distance formula to develop techniques for estimating the area under a curve.

The Distance Formula:

The distance between two points on the same horizontal or vertical line can be found by taking the difference of the x or y coordinates.

Segment AB and segment BC can found by using the above rule. By taking the differences in the x and y coordinates segment Ab is 4 units and segment BC is 3 units. Segment AC is the hypotenuse of the triangle. It can be found by utilizing the Pythagorean Theorem.

Segment AC can also be found using the same approach used on segments AB and BC.

To prove for any triangle:

Given: Poits A and B with coordinates (x1, y1) and (x2, y2).

Therefore the distance formula is as follows:

In a coordinate plane the distance ,d, between two points (x1, x2) and (y1, y2) is given by d = sqrt[(x2-x1)^2 + (y2-y1)^2].

Quadrature is a procedure of approximating the area of an enclosed region on a plane by using the sum of the areas of a number of rectangles.

I. Left-Hand Rule:

* Start with a circle with radius of 5 units. Draw rectangles with the upper left vertex of each rectangle touching the curve.

* Find the y coordinates for each vertex. by drawing a line connecting the origin of the circle (0,0) to each vertex. This forms a series of right triangles with thr radius being the hypotenuse where r = 5 units. The basae of the triangle is the x coordinate, so the third side can be found tusing the Pytthagorean Theorem. y^2 = r^2 - x^2.

* Find the area of each rectangle. Note: the base of each rectangle is 1.

* Find the sum of the areas of the rectangles by completing the pattern. This will give you an estimation.

* Multiply the estimation by four to get an estimation of the area of the original circle. 4 X 21.481555 = 85.92622. Check this by checking area of the circle with GSP.

* You can see by GSP that this method overestimates the area of the cicle and curve. The overestimation can be seen by looking at the area of the rectangles used. The rectangles use more area than the circle has.

* To find relative error: E = [abs(Ve - Vt)]/Vt * 100, where Ve = estimated value, Vt = true value, and E = percent of error. , so the Percent of error is 8.99%.

II. Right Hand Rule:

* Start with the same circle with radius 5 units as in the left hand rule. Draw the rectangles so that the upper right vertex is touching the circle.

* Complete similiar method as left hand rule. Estimate area.

* This method understimates the area because the estimated area obtained was 65.92622 and the true area is 78.84. Now calculate the relative error..

III. Average areas from parts I and II:

Take the average area of parts I and II to get a new area.

Now find the relative error for the average of areas.

By averaging the two areas, the percent of error decreases.