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Numerical integration
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Numerical integration
Lessons
Here are the three following ways to estimate the value of a definite integral with n sub-intervals:
1) Midpoint Rule
Mn=∫abf(x)dx≈Δx[f(x1)+f(x2)+...+f(xn−1)+f(xn)]
Where xi is the midpoint of each interval.
2) Trapezoid Rule
Tn=∫abf(x)dx≈2Δx[f(x0)+2f(x1)+2f(x2)+...+2f(xn−1)+f(xn)]
3) Simpsons Rule
Sn=∫abf(x)dx≈3Δx[f(x0)+4f(x1)+2f(x2)+...+2f(xn−2)+4f(xn−1)+f(xn)]
If f′′ is continuous from [a,b] and there is a value M such that ∣f′′(x)∣≤M for all x∈[a,b], then we can use the following formulas to calculate the error of the Midpoint and Trapezoid Rule:
1) Midpoint Rule Error Formula
EM≤24n2M(b−a)3
2) Trapezoid Rule Error Formula
ET≤12n2M(b−a)3
If f(4)(x) is continuous from [a,b] and there is a value K such that ∣f(4)(x)∣≤K for all x∈[a,b], then we can use the following formulas to calculate the error of Simpsons Rule:
3) Simpson's Rule Error Formula
ES≤180n4K(b−a)5
Here is a formula that may be of use when calculating the points of interest in Trapezoid and Simpsons Rule:
xi=a+iΔx
Where xi is the point of interest at i.
- IntroductionOverview of Numerical Integration
- Midpoint, Trapezoid and Simpsons Rule
- 1.Questions Regarding the Midpoint Rule
Approximate ∫49xdx using Midpoint Rule with 5 sub-intervals.
- 2.Approximate ∫252+x21 using Midpoint Rule with 3 sub-intervals.
- 3.Questions Regarding the Trapezoid Rule
Approximate ∫01exdx using Trapezoid Rule with 4 sub-intervals.
- 4.Approximate ∫15x2dx using Trapezoid Rule with 5 sub-intervals.
- 5.Questions Regarding the Simpsons Rule
Approximate ∫24x−2dx using Simpsons Rule with 4 sub-intervals.
- 6.Approximate ∫14ln(x2)dx using Simpsons Rule with 6 sub-intervals.
- 7.Questions Regarding Error Bounds
Let f(x)=ex3 consider ∫01ex3dx. Assume you know that ∣f′′(x)∣≤15e and ∣f(4)∣≤585e for all x∈[0,1]. If n = 10, then find the following errors:
a)EMb)ETc)ES