Question

Use n=6 subdivision to approximate the value of
∫_0^6▒√(x+2) dx by the trapezoidal rule

Answer 1

$∫_0^6\sqrt{(x+2)}$

We divide the segment [0;6] into 6 parts and at each node we calculate the value of the integrand

$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c:c:c:} n & 0 & 1&2&3&4&5&6\\ \hline x_n & 0 & 1&2&3&4&5&6\\ \hdashline \approx\sqrt{x+2} &1.41&1.73&2&2.24&2.45&2.65&2.83 \end{array}$

The formula for calculating the integral by the trapezoid method

$∫_a^bf(x)dx\approx{h}[\frac{f(x_0)+f(x_n)}{2}+f(x_{1})+...+f(x_{n-1})]\ where \ h =\frac{(b-a)}{n}$

$h=\frac{6-0}{6}=1$

$∫_0^6\sqrt{(x+2)}\approx\frac{1.41+2.83}{2}+1.73+2.0+2.24+2.45+2.65 =13.19$

Answer: $∫_0^6\sqrt{(x+2)}\approx13.19$