Consider the function y=f(x) for the interval a≤x≤b, shown in figure:
To evaluate the definite integral, dx, we divide the interval a≤x≤b into subintervals each of length. Then, the number of points between x0=a,and xn=b is x1=a+∆x,x2=a+2∆x,…xn-1=a+(n-1)∆x. Therefore, the integral from a to b is the sum of the integrals from a to x1, from x1 to x2 and so on, and finally from xn-1 to b.
The total area is:
0 P1 x1that is equal to plus the area of the trapezoid x1 P1 P2 x2 that is equal to , and so on. Then, the trapezoidal approximation becomes
Using the trapezoidal rule with n=4, estimate the cost of the definite integral
Compare with the exact value and evaluate the percent error.
The exact value of this integral is
For the trapezoidal rule approximation, we have
and by substitution into an equation
From equation 3 and equation 4, we find that the percent error is