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?

1 Answer
Alan P.
May 28, 2015

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

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