## Tuesday, June 4, 2013

### 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.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

## Thursday, May 24, 2012

### Natural Logarithm

Some book authors define the natural logarithm function, ln (x) as

$ln(x)=\int_{1}^{x}\frac{1}{t}dt,\; \; x>0$.

If we take a look at the graphs below, the red curve represents the graph of $y=\frac{1}{x}$, and the dark green curve represents the graph of $y=ln(x)$. The purple region falls below the red curve is bounded between x = h and 1, while the blue region is bounded between x = 1 and k.

Now scroll the value of k, do you notice that point K moves according to the k value? The y-coordinate of point K is the area of the blue region. Same concept applies to the value of h and point H. Here comes the question, do you know how to get negative y-coordinate for point H?

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Betty, Created with GeoGebra