Conic Sections

Conic section is the intersection between a cone and a plane. To make the explanation simple, I use an upright cone for the following examples. There are basically three types of conic sections:

1. Ellipse
2. Parabola
3. Hyperbola

Let's take a look at how the conic sections are formed:

1. Ellipse
   An ellipse is formed when a slanted plane ax + by + cz = d intersects with the upright cone. Take note the angle of the plane from the z-axis is bigger than the angle of the cone from the z-axis. Otherwise, a hyperbola is form.
   When a, b = 0, the plane becomes a horizontal plane, cz = d. This makes the curve of intersection changes from the ellipse to a circle.
2. Parabola
   A parabola is formed when a plane that is "parallel" to the cone intersects with the cone as shown in the first graph below.
3. Hyperbola
   When a plane ax + by + cz = d (slanted/vertical) that forms an angle from the z-axis is smaller than the angle of the cone from the z-axis (as mentioned in ellipse just now), a hyperbola is formed.
   

Betty, Created with GeoGebra

Midpoint Rule

Some people estimate the area below the curve using Midpoint Rule. What is Midpoint Rule?

We'll use the graph below to explain about the concept. From the graph, we can see that the region falls between graph f, x-axis and y-axis. The interval of x is from 0 to 1.

Let's say we use one rectangle to estimate the area of the region. This rectangle starts from x = 0 to x = 1. The middle x is 0.5. We take the y value when x = 0.5 as the height of the rectangle. Thus, a rectangle is formed and the area of the rectangle is 0.75.

Now, drag the value of n to 2, you will see that there are two rectangles now. For the first rectangle, x is from 0 to 0.5; for the second rectangle, 0.5 to 1. Take a look at the first rectangle, the height of the first rectangle is the y value when x = 0.25, which is the middle x between 0 and 0.5; while the second rectangle take the height using the y value when x = 0.75, which is the middle x between 0.5 and 1. Do you realize that the approximation of the area is nearer to the exact area?

Now, try to slide the value of n and see the changes in the approximation. You will see that the approximation is getting more accurate as the value of n increases.

Betty, Created with GeoGebra