MAT 1235 Calculus II Section 7.7 Approximate (Numerical) Integration

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Presentation transcript:

MAT 1235 Calculus II Section 7.7 Approximate (Numerical) Integration

HW and … No WebAssign Do problem # (c). No need to turn in. Answers 20) 83, 59 22) 20

Idea Many useful Integrals are difficult/impossible to evaluate by the Fundamental Theorem of Calculus

Probability & Statistics

Idea In practice, we estimate the values of these integrals.

Method I: Riemann Sum

Method II: Trapezoidal Rule

Method III: Simpson’s Rule

Even Number of Intervals

Facts In the last lab, we learn different methods of estimating the value of definite integrals. In general, given a method, bigger n gives better approximation. We want to find the smallest n such that |Error|<certain accuracy Why?

Facts In the last lab, we learn different methods of estimating the value of definite integrals. In general, given a method, bigger n gives better approximation. We want to find the smallest n such that |Error|<certain accuracy In general, we do not know the error. Why?

Rules Midpoint Rule Rectangles Height of rectangle = function value of the midpoint of each subinterval Trapezoidal Rule Trapezoids Simpson’s Rule Parabolas n = even

Error Bounds - Trapezoidal Rule

Example 1 How large should we take n in order to guarantee that the trapezoidal rule approximation for is accurate to within ?

Example 1: Analysis We want to find the smallest n such that If so, then

Example 1: Analysis Once we find K, n is computed by solving the inequality

Example 1: Find K We need to find the abs. max. of on [1,2]

Abs. Max. of |h(x)| FACT: The abs. max. of |h(x)| occurs at the abs. max. or min. of h(x).

Modified Closed Interval Method Abs. max. of |h(x)| occurs at the end points or critical numbers of h(x).

Step 1: Find K Let, then

Step 2: Find n

Example 1: Q: Can we choose a bigger K?

Remarks We are solving for inequalities, not equations The n here is not necessarily the actual minimum. It is the minimum guaranteed by the error formula. Error bounds for midpoint and Simpson’s rule are similar. YES, you need to know the formulas for quizzes and exams.