This question has four part to, please try and explain your answer for me to get

Question

This question has four part to, please try and explain your answer for me to get a better understanding

Directions: For the function ?_0^2??f(x)=(x-2)^2+2 ? dx we are going to estimate the area under the curve using the trapezoidal rule where n = 5 by doing the following:

Part 1 Divide the interval into 5 equal pieces. How long is each piece? This will represent the width of the rectangles we will use to estimate the area in gray above. What will the x-values be for the endpoint of each piece?
Width of rectangle = ?x=
x_1=
x_2=
x_3=
x_4=
x_5=
x_6=


Part 2 Evaluate f(x) for x1 to x6 the endpoints. This will represent the bases of the trapezoids we will use to estimate the area under the curve.
f(x_1 )=
f(x_2 )=
f(x_3 )=
f(x_4 )=
f(x_5 )=
f(x_6 )=



part 3 Using the heights from part 2 and the width from part 1 draw the trapezoids onto the graph. Be sure that you measure the height from each endpoint.

Based on the sketch will the estimate you get be an over or underestimate? Why?


part 4 The Trapezoidal rule is ?_a^b??f(x)dx??x/2(f(x_1)+2f(x_2 )+?+2f(x_(n-1) )+f(x_n))?. Plug the values you got in part II into this formula. (Hint: you should have 5 terms inside the parenthesis with the middle three terms all multipled by 2.) What estimate for the definite integral ?_0^2??f(x)=(x-2)^2+2 ? dx did you get?

Explanation / Answer

a.) interval= (0,2) so 5 equal divisions give length of 1 division = 0.4 x1= 0 x2= 0.4 x3= 0.8 x4= 1.2 x5= 1.6 x6= 2 b.) f(x1) =6 f(x2)= 4.56 f(x3)= 3.44 f(x4)= 2.64 f(x5)= 2.16 f(x6)= 2 this gives a less area than integration as some part of the curve is left. d.) (2-0)/10* (6+2*(4.56)+2*(3.44)+2*(2.64)+2*(2.16)+2) = 2/10 *33.6 =67.2/10 = 6.72

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