Question

State trapezoidal rule for numerical integration.

Answer 1

Trapezoidal Rule Formula. In mathematics, and more specifically in numerical analysis, the trapezoidal rule, also known as the trapezoid rule or trapezium rule, is a technique for approximating the definite integral. ... The trapezoidal rule is to find the exact value of a definite integral using a numerical method.

Answer 2

You can also approximate the value of a definite integral by using a trapezoid rather than a rectangle. The area under the curve is approximated as a trapezoid rather than a rectangle.

The trapezoidal rule for evaluating definite integrals uses a non-rectangular trapezoid to approximate the area under the curve. To get an idea of ​​the final shape of the rule, consider the trapezoid.

Assume that the length of each subinterval is given by Δx.

First, recall that the area of ​​a trapezoid with height h and base lengths b1(

) and b2(

) is defined by Area=12h(b1+b2).

We see that the first trapezoid has height Δx

and parallel bases with lengths f(x0)

and f(x1).

So the area of ​​the first trapezoid  is equal to

12∆x(f(x0)+f(x1)).

The areas of the remaining three trapezoids are

12Δx(f(x1)+f(x2)), 12Δx(f(x2)+f(x3)), and 12Δx(f(x3)+f(x4)).

So

∫baf(x)dx≈12∆x(f(x0)+f(x1))+12∆x(f(x1)+f(x2))+12∆x(f(x2) + f(x3)) +12∆x (f(x3)+f(x4)). Eliminating the

common divisor 12Δx

and combining similar terms yields

∫baf(x)dx≈12Δx(f(x0)+2f(x1)+2f(x2)+2f(x3)+f(x4) ).

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