(1)Trapezoidal rule :
where h =
.
here, n = 8, since we are dividing the interval into 8 equal parts.
So, a = x0 = 0, x1 = 1/8=0.125,x2 = 0.25, x3 = 0.375, x4= 0.5, x5= 0.625, x6= 0.75, x7= 0.875, x8= b = 1.
xi f(xi)=e2xi
(2)
Simpson's rule : Let 2m = 8. h=
=
.
=
[1 + 4(e
1/4+e
3/4+e
5/4+e
7/4)+ 2(e
1/2 + e + e
3/2) + e
2 ].
=
[1 +4(1.284 + 2.117 + 3.4903 +5.7546) + 2(1.6487 + 2.7182 + 4.4816) + 7.389 ]
=
[1 + 50.5836 + 17.697 + 7.389 ]
= 3.1529.
It's actual value is :
(1) prcentage error from it's true value in trapezoidal case :
Trapezoidal value - actual value = 3.2111 - 2.6945 = .5166.
Percentage error from it's true value =
* 100 %= 16.08%.
∴ 16% error from it's true value.
(2) Percentage error from it's true value in simpson's rule :
Simpson's value - actual value = 3.1529 - 2.6945 = 0.4584.
Percentage error from it's actual value =
*100% = 17.01%
∴ 17.01 % error from it's true value.