Trapezoidal Approximation

Trapezoidal ApproximationObjective: To find area using trapezoids.

Trapezoidal ApproximationWe will now approximate the area under a curve by using trapezoids rather than rectangles. This is only an approximation; we will never take the limit to find the exact area.

Trapezoidal ApproximationAs we did before, we will draw vertical lines to divide the interval into n subintervals. This time, we will construct n trapezoids.

Trapezoidal ApproximationAs we did before, we will draw vertical lines to divide the interval into n subintervals. This time, we will construct n trapezoids.The area of a trapezoid is . The height of the each trapezoid is what we called before.

Trapezoidal ApproximationThe area of the first trapezoid is

The area of the second trapezoid is

The area of the third trapezoid is

Trapezoidal ApproximationThe area of the first trapezoid is

The area of the second trapezoid is

This leads us to the following:

Sample AP Question

Sample AP Question

NoteThe AP book made the following notes:DO NOT use your calculators statistics regression equation to find f(x) and then the Fundamental Theorem of Calculus; this may give you a more accurate answer (D) but it is not what you were asked for. The left hand sum is (E) and the right hand sum is (A).

AP QuestionUsing the subintervals [1,5], [5,8], and [8,10], what is the trapezoidal approximation to

AP QuestionUsing the subintervals [1,5], [5,8], and [8,10], what is the trapezoidal approximation to

AP QuestionUsing the subintervals [1,5], [5,8], and [8,10], what is the trapezoidal approximation to

AP QuestionThe expression is the trapezoidal approximation for which of the following definite integrals?

a) b) c)

d) e)

AP QuestionThe expression is the trapezoidal approximation for which of the following definite integrals?There are 4 trapezoids (why), so n = 4.We know that and if n = 4 it becomes so b a = 2. Our only two choices are A and D.

AP QuestionThe expression is the trapezoidal approximation for which of the following definite integrals?There are 4 trapezoids (why), so n = 4.We know that and if n = 4 it becomes so b a = 2. Our only two choices are A and D.If a = 0, the bases are f(0), f(1/2), f(1), f(3/2), f(2).If a = 1, the bases are f(1), f(3/2), f(2), f(5/2), f(3).

AP QuestionThe expression is the trapezoidal approximation for which of the following definite integrals?There are 4 trapezoids (why), so n = 4.We know that and if n = 4 it becomes so b a = 2. Our only two choices are A and D.If a = 0, the bases are f(0), f(1/2), f(1), f(3/2), f(2).If a = 1, the bases are f(1), f(3/2), f(2), f(5/2), f(3).

AP QuestionThe expression is the trapezoidal approximation for which of the following definite integrals?

a) b) c)

d) e)

AP QuestionThe expression is the trapezoidal approximation for which of the following definite integrals?

a) b) c)

d) e)

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