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

1 Answer
Alan P.
Jul 14, 2015

Evaluate #y=x^2-2x# for #x epsilon {0, 1, 2, 3}#; 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
#color(white)("XXXX")##= "width" xx ("height at " x_1 + "height at " x_2)/2#
#color(white)("XXXX")##color(white)("XXXX")#(in this case the #"width" = 1# for all trapezoids).

#"Area"_(0:1) = 1 xx(1+0)/2 = 1/2#

#"Area"_(1:2) = 1xx(0+1)/2 = 1/2#

#"Area"_(2:3) = 1xx (1+4)/2 = 2 1/2#

Total Area of the 3 Trapezoids (which approximate area under the curve)
#color(white)("XXXX")##= 1/2 + 1/2 + 2 1/2 = 3 1/2#

#color(white)("XXXX")##color(white)("XXXX")##...that's #3+1/2# or #3.5# for our friends in France who have never seen mixed fractions before ; )

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