Trapezoidal Rule Integration

The Trapezoidal rule integration is an approximate technique for calculating the definite integral. The procedure of measuring the area under a function plotting on a graph.

int_a^bf(x)dx

Here,

a = lower limit of integration

b = upper limit of integration

The trapezoidal rule act as approximating the region under becomes that,

int_a^bf(x)dx  ~~ (b-a) (f(a)+f(b))/2

Basis of Trapezoidal Rule:

Trapezoidal Rule is state’s approximate of the integrand as an nth term polynomial,

I =  int_a^bf(x)dx

Where f(x) = fn(x)

and  fn(x)= a0+a1x+….a n-1xn-1+anxn

Overview

Then the integral function is approximated by the integral of that nth term polynomial.

int_a^bf(x)dx    ~~ int_a^bf(x)dx

Trapezoidal Rule integration plugging n=1, that the area under the linear polynomial,

int_a^bf(x)dx = (b-a) (f(a)+f(b))/2

Method Derived From Geometry:

The area under the curve is a trapezoid. The integral

int_a^bf(x)dx  = Area of trapezoid

= half (sum parallel sides) (height)

=   (f(b)+f(a)(b-a))/2

= ((b-a)f(a)+f(b))/2


Exercise: Using n = 5, approximate

int_0^1sqrt(x^2+1)dx


Here, a = 0 and b = 1

Deltax

= (b-a)/n = (1-0)/5 = 0.5

y0 = f(a) = f(0) = sqrt(0^2+1) = 1

y1 = f(a + Δx) = f(0.2) = sqrt((0.2)^2+1) = 1.019

y2 = f(a + 2Δx) = f(0.4) =sqrt(0.4^2+1)

y3 = f(a + 3Δx) = f(0.6) =sqrt(0.6^2+1)

y4 = f(a + 4Δx) = f(0.8) =sqrt(0.8^2+1)

y5 = f(b) = f(1) =sqrt(1^2+1)

So the area ≈ 0.2( 1/2 *1.019+1.077+1.166+1.128... 1/2 *1.414)

So int_0^1sqrt(x^2+1)dx ≈1.150

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The error of the integration trapezoidal rule is the difference between the value of the integral and the numerical solution

Error= int_a^bf(x)dx - (b-a)/n ((f(a)+f(b))/2 + sum_(k=1)^(n=1)(a+k (b-a)/n)

This error can be written as,

Error= - ((b-a)^3/(12n)^2) f''(ξ)

where ξ is some numbers between a and b.

If the integrand is concave up and it has a positive 2nd derivative, then the error is negative and the trapezoidal rule overestimates the true value. It can also be seen from the geometric figure. The trapezoids regard the area under the curve and extend over it. Same as a concave down function yields an underestimate. Because area is unaccounted for under curve. If the interval of the integral stay approximated admit an inflection point, then the error is very difficult to identify.

an asymptotic error estimate for n → ∞ is given by

Error =   -   ((b-a)^2)/(12n^2) [f'(b)-f'(a)]+O(n-3)

It is disagree that the speed of convergence of the trapezoidal rule reflects and can be used as a definition of classes of smoothness of the functions.