# How do you use the trapezoidal rule with n=2 to approximate the area under the curve y=1/x^2 from 1 to 3?

Jul 15, 2015

Draw 2 trapezoids under the curve, the first between $x = 1$ and $x = 2$ and the second between $x = 2$ and $x = 3$; use the sum of the areas to approximate the required area.

#### Explanation:

With $n = 2$ we have 2 trapezoids
The first will have a width of $1$ (the distance on the X-axis between 1 and 2)
and an average height of $\frac{\frac{1}{1} ^ 2 + \frac{1}{2} ^ 2}{2} = \frac{5}{8}$

The second will also have a width of $1$ (the distance between 2 and 3)
and will have an average height of $\frac{\frac{1}{2} ^ 2 + \frac{1}{3} ^ 2}{2} = \frac{13}{36}$

The total area of the two trapezoids (approximating the area under the curve) is
$\textcolor{w h i t e}{\text{XXXX}}$$\frac{5}{8} + \frac{13}{36} = \frac{45 + 26}{72} = \frac{71}{72}$