Estimate the integral of sqrt (1 + ln xdx) with superscript of 2 and subscript of 1 outside the bracket by usi

Question

Estimate the integral of sqrt (1 + ln xdx) with superscript of 2 and subscript of 1 outside the bracket by using
(a) The trapezoidal rule
(b) Simpson’s Rule.

Answer 1

Hi,

a) The trapezoidal rule states that
[b lim a]∫f(x)dx = [(b -a)/2][f(a) + f(b)] ......(1)

If you want more accurate results divide the interval between a and b to around say 10 intervals, like 1 to 1.1, 1.1 to 1.2, ...............1.9 to 2.

and evaluate RHS of eqn(1) for each smaller a, b and finally add all of them to get the integral


b) Simpson rule states that
[b lim a]∫f(x)dx = [(b -a)/6][f(a) + 4f(0.5a +0.5b) + f(b)]

for more accuracy divide the overall interval to small intervals and follow the same procedure as shown under trapezoid rule.

Good luck

Answer 2

Using 0.1 intervals,
. x . . . f(x) . . . . Mean . . . Trap. . Simpson's
1.00 1.00000
1.05 1.02410 0.10241 0.10233 0.10238
1.10 1.04657
1.15 1.06760 0.10676 0.10670 0.10674
1.20 1.08735
1.25 1.10596 0.11060 0.11054 0.11058
1.30 1.12355
1.35 1.14022 0.11402 0.11398 0.11401
1.40 1.15606
1.45 1.17114 0.11711 0.11708 0.11710
1.50 1.18552
1.55 1.19927 0.11993 0.11990 0.11992
1.60 1.21244
1.65 1.22506 0.12251 0.12248 0.12250
1.70 1.23719
1.75 1.24885 0.12488 0.12486 0.12488
1.80 1.26007
1.85 1.27090 0.12709 0.12707 0.12708
1.90 1.28135
1.95 1.29144 0.12914 0.12913 0.12914
2.00 1.30121
Area . . . . . . .. 1.17445 1.17407 1.17433