# How do you use the trapezoid rule to approximate the equation y=x^2 -2x + bounded by y=0, x=0, and x=3?

Jul 14, 2015

Evaluate $y = {x}^{2} - 2 x$ for $x \epsilon \left\{0 , 1 , 2 , 3\right\}$; calculate the areas between adjacent $x$ values and sum the areas to get an approximation (3.5 sq. units)

#### Explanation:

(See diagram of the function graph and approximating trapezoids below).

Area of each trapezoid
$\textcolor{w h i t e}{\text{XXXX}}$= "width" xx ("height at " x_1 + "height at " x_2)/2
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$(in this case the $\text{width} = 1$ for all trapezoids).

${\text{Area}}_{0 : 1} = 1 \times \frac{1 + 0}{2} = \frac{1}{2}$

${\text{Area}}_{1 : 2} = 1 \times \frac{0 + 1}{2} = \frac{1}{2}$

${\text{Area}}_{2 : 3} = 1 \times \frac{1 + 4}{2} = 2 \frac{1}{2}$

Total Area of the 3 Trapezoids (which approximate area under the curve)
$\textcolor{w h i t e}{\text{XXXX}}$$= \frac{1}{2} + \frac{1}{2} + 2 \frac{1}{2} = 3 \frac{1}{2}$

$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$$\ldots t \hat{'} s$3+1/2$\mathmr{and}$3.5# for our friends in France who have never seen mixed fractions before ; )