Numerical integration

Numerical integration

Basic concepts: Riemann sum,

Lessons

Here are the three following ways to estimate the value of a definite integral with nn sub-intervals:

1) Midpoint Rule

Mn=abf(x)dxΔx[f(x1)+f(x2)+...+f(xn1)+f(xn)]M_{n} = \int^b_a f(x)dx \approx \Delta x[f(x_{1})+f(x_{2})+...+f(x_{n-1})+f(x_{n})]

Where xix_{i} is the midpoint of each interval.

2) Trapezoid Rule

Tn=abf(x)dxΔx2[f(x0)+2f(x1)+2f(x2)+...+2f(xn1)+f(xn)]T_{n} = \int^b_a f(x)dx \approx \frac{\Delta x}{2} [f(x_{0})+2f(x_{1})+2f(x_{2})+...+2f(x_{n-1})+f(x_{n})]

3) Simpsons Rule

Sn=abf(x)dxΔx3[f(x0)+4f(x1)+2f(x2)+...+2f(xn2)+4f(xn1)+f(xn)]S_{n} = \int^b_a f(x)dx \approx \frac{\Delta x}{3} [f(x_{0})+4f(x_{1})+2f(x_{2})+...+2f(x_{n-2})+4f(x_{n-1})+f(x_{n})]

If ff'' is continuous from [a,ba, b] and there is a value MM such that f(x)M|f''(x)| \leq M for all x[a,b]x \in [a, b], then we can use the following formulas to calculate the error of the Midpoint and Trapezoid Rule:

1) Midpoint Rule Error Formula

EMM(ba)324n2E_{M} \leq \frac{M(b-a)^{3}}{24n^{2}}

2) Trapezoid Rule Error Formula

ETM(ba)312n2E_{T} \leq \frac{M(b-a)^{3}}{12n^{2}}

If f(4)(x)f^{(4)} (x) is continuous from [a,b][a, b] and there is a value KK such that f(4)(x)K|f^{(4)} (x)| \leq K for all x[a,b]x \in [a, b], then we can use the following formulas to calculate the error of Simpsons Rule:

3) Simpson’s Rule Error Formula

ESK(ba)5180n4E_{S} \leq \frac{K(b-a)^{5}}{180n^{4}}

Here is a formula that may be of use when calculating the points of interest in Trapezoid and Simpsons Rule:

xi=a+iΔxx_{i} = a + i\Delta x

Where xix_{i} is the point of interest at ii.

  • Introduction
    Overview of Numerical Integration

    - Midpoint, Trapezoid and Simpsons Rule


  • 1.
    Questions Regarding the Midpoint Rule

    Approximate 49xdx\int^9_4 \sqrt{x} dx using Midpoint Rule with 5 sub-intervals.


  • 2.
    Approximate 2512+x2\int^5_2 \frac{1}{2+x^{2}} using Midpoint Rule with 3 sub-intervals.

  • 3.
    Questions Regarding the Trapezoid Rule

    Approximate 01exdx\int^1_0 e^{x} dx using Trapezoid Rule with 4 sub-intervals.


  • 4.
    Approximate 15x2dx\int^5_1 x^{2} dx using Trapezoid Rule with 5 sub-intervals.

  • 5.
    Questions Regarding the Simpsons Rule

    Approximate 24x2dx\int^4_2 \sqrt{x-2} dx using Simpsons Rule with 4 sub-intervals.


  • 6.
    Approximate 14ln(x2)dx\int^4_1 \ln (x^{2}) dx using Simpsons Rule with 6 sub-intervals.

  • 7.
    Questions Regarding Error Bounds

    Let f(x)=ex3f(x) = e^{x^{3}} consider 01ex3dx\int^1_0 e^{x^{3}} dx. Assume you know that f(x)15e|f''(x)| \leq 15e and f(4)585e|f^{(4)}| \leq 585e for all x[0,1]x \in [0, 1]. If nn = 10, then find the following errors:

    a)
    EME_{M}

    b)
    ETE_{T}

    c)
    ESE_{S}