Question

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?

Answer 1

With `n=6` over the interval `[-3,+3]`
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) (f(x)+f(x+1))/2xx(1)`
or
`(f(-3)+f(+3))/2 + sum_(x=-2)^(x=+2) f(x)`

#{: ( 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
`(-35)`

Note this area is negative because the entire curve between `[-3,+3]` is below the X-axis