UPSC 2014 Maths Optional Paper 2 Q5c — Step-by-Step Solution

10 marks · Section B

Trapezoidal rule (composite; error) · Numerical Analysis · asked 3× in 13 yrs · Read the full method →

Question

Use five subintervals to integrate $\displaystyle\int_0^1\dfrac{dx}{1+x^{2}}$ using trapezoidal rule.

Composite trapezoidal rule with equal subintervals.

Setup. $h=\dfrac{1-0}{5}=0.2$ . Points $x_i=0,\,0.2,\,0.4,\,0.6,\,0.8,\,1.0$ .

T=\dfrac{h}{2}\bigl[f_0+f_5+2(f_1+f_2+f_3+f_4)\bigr].

Sum of interior values: $f_1+f_2+f_3+f_4=0.961538+0.862069+0.735294+0.609756=3.168657$ .

T=\dfrac{0.2}{2}\bigl[1.000000+0.500000+2(3.168657)\bigr]=0.1\cdot[1.500000+6.337314]=0.1\cdot 7.837314=0.7837.

\boxed{\;\int_0^1\dfrac{dx}{1+x^{2}}\approx 0.7837.\;}