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Trapezoidal Approximation

Trapezoidal Approximation. Objective: To find area using trapezoids. Trapezoidal Approximation. We 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.

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Trapezoidal Approximation

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  1. Trapezoidal Approximation Objective: To find area using trapezoids.

  2. Trapezoidal Approximation • We 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.

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

  4. Trapezoidal Approximation • As 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.

  5. Trapezoidal Approximation • The area of the first trapezoid is • The area of the second trapezoid is • The area of the third trapezoid is

  6. Trapezoidal Approximation • The area of the first trapezoid is • The area of the second trapezoid is • This leads us to the following:

  7. Sample AP Question

  8. Sample AP Question

  9. Note • The AP book made the following notes: • DO NOT use your calculator’s 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).

  10. AP Question • Using the subintervals [1,5], [5,8], and [8,10], what is the trapezoidal approximation to

  11. AP Question • Using the subintervals [1,5], [5,8], and [8,10], what is the trapezoidal approximation to

  12. AP Question • Using the subintervals [1,5], [5,8], and [8,10], what is the trapezoidal approximation to

  13. AP Question • The expression is the trapezoidal approximation for which of the following definite integrals? a) b) c) d) e)

  14. AP Question • The 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.

  15. AP Question • The 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).

  16. AP Question • The 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).

  17. AP Question • The expression is the trapezoidal approximation for which of the following definite integrals? a) b) c) d) e)

  18. AP Question • The expression is the trapezoidal approximation for which of the following definite integrals? a) b) c) d) e)

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