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 ; )

enter image source here

Related questions
See all questions in Integration Using the Trapezoidal Rule
Impact of this question
4667 views around the world
You can reuse this answer
Creative Commons License