# How do you estimate the area under the curve f(x)=x^2-9 in the interval [-3, 3] with n = 6 using the trapezoidal rule?

May 28, 2015

With $n = 6$ over the interval $\left[- 3 , + 3\right]$
we have 6 trapezoids each with a width of $1$ unit

The Sum of the Areas of these trapezoids is
${\sum}_{x = - 3}^{x = + 2} \frac{f \left(x\right) + f \left(x + 1\right)}{2} \times \left(1\right)$
or
$\frac{f \left(- 3\right) + f \left(+ 3\right)}{2} + {\sum}_{x = - 2}^{x = + 2} f \left(x\right)$

{: ( x, color(white)("xxxxxx"), f(x)=x^2-9), (-3, color(white)("xxxxxx"), 0), (-2, color(white)("xxxxxx"), -5), (-1, color(white)("xxxxxx"), -8), ( +0, color(white)("xxxxxx"), -9), (+1, color(white)("xxxxxx"), -8), (+2, color(white)("xxxxxx"), -5), (+3 color(white)("xxxxxx"),, 0) :}

and our estimated area under the curve is
$\left(- 35\right)$

Note this area is negative because the entire curve between $\left[- 3 , + 3\right]$ is below the X-axis