Sect 7 7 Approximate Integration Review from Calc

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Sect. 7. 7 Approximate Integration Review from Calc I – Fundamental Theorem of Calculus

Sect. 7. 7 Approximate Integration Review from Calc I – Fundamental Theorem of Calculus f(x) is continuous on [a, b] F is an antiderivative of f an antiderivative F can’t be found If (1)___________________________, or (2)_________________________________ a function is determined by collected data we can use approximation techniques to estimate definite integrals _________________________________ Techniques Use Riemann Sums using (1) _________________________ left endpoints * ______________ right endpoints * ______________ midpoints * ______________ Trapezoidal Rule (2) _____________ (3) _____________ Simpson’s Rule

Riemann Sums

Riemann Sums

Trapezoid Rule For n = 4 trapezoids f(x 1) f(x 2) f(x 3) f(x

Trapezoid Rule For n = 4 trapezoids f(x 1) f(x 2) f(x 3) f(x 0) A 2 A 3 A 4 A 1 f(x 4) + _____________ a b

 Use for Midpt Rule Use for Trap Rule

Use for Midpt Rule Use for Trap Rule

 Observations: Drawing in the tangent line BC at point P makes _______________________ the

Observations: Drawing in the tangent line BC at point P makes _______________________ the two right triangles BMP and CNP equal in _______________________ area, so the rectangle, AMND, formed by the _______________________ midpoint is the same area as the trapezoid, ______________________________________________ ABCD, with the tangent line BC. The trapezoid, AQRD, is formed by the trapezoid _______________________ method. _______________________ *Errors are always opposite in sign. _______________________ *Size of the error in the Midpoint Rule is about _______________________ half the size of the Trapezoid Rule. _______________________ *Blue shading represents error in Trapezoid Rule. _______________________ *Red shading represents error in Midpoint Rule. _______________________

 Error Bounds: Midpoint Rule: ________________________ Trapezoidal Rule: ________________________ Calculate the Error Bound for

Error Bounds: Midpoint Rule: ________________________ Trapezoidal Rule: ________________________ Calculate the Error Bound for Midpoint and the Error Bound for Trapezoidal

Simpson’s Rule: Subintervals grouped in pairs…so n must be and even integer. * ______________________________________

Simpson’s Rule: Subintervals grouped in pairs…so n must be and even integer. * ______________________________________ Uses parabolas passing through 3 points on the pair of subintervals. * ______________________________________ Formula: _________________________________ Error Bound for Simpson’s Rule: _________________________