Question

Use (a) trapezoidal Rule, (b) The midpoint rule, and (c) Simpson’s Rule withto approximate the given integral. Round your answer to six decimal places

Short answer

1. Trapezoidal Rule ofis 0.641245
2. Midpoint Rule of0.642734
3. Simpson’s Rule of 0.642551

Step-by-step solution

Step 1: Solution of applied Trapezoidal Rule

By using trapezoidal rule

\(\begin{aligned}{l}f(x) = \sqrt x \cos xdx\\\Delta x = \frac{{b - a}}{n}\\\frac{{4 - 1}}{{10}} = 0.3\\{T_{10}} = (f(0) + 2f(0.3) + 2f(0.6) + \cdots + 2f(1.2) + f(2))\frac{{\Delta x}}{2}\\ \approx 0.641245\end{aligned}\)

Step 2: Solution of applied midpoint Rule

Let, consider the interval, there are 10 subintervals to consider

\(\begin{aligned}{l}Mi{d_6} = \Delta x\left( {f\left( {{m_1}} \right) + f\left( {{m_2}} \right) + f\left( {{m_3}} \right) + f\left( {{m_4}} \right) + f\left( {{m_5}} \right) + f\left( {{m_6}} \right)} \right)\\ = \Delta x\left( {f\left( {1.25} \right) + f\left( {1.75} \right) + f\left( {2.25} \right) + f\left( {2.75} \right) + f\left( {3.25} \right) + f\left( {3.75} \right)} \right)\\ \approx 0.642734\end{aligned}\)

Step 3: Solution of applied Simpson’s Rule

Simpson’s Rule

\(\begin{aligned}{l}f(x) = \sqrt x \cos xdx\\\Delta x = \frac{{b - a}}{n}\\\frac{{4 - 1}}{{10}} = 0.3\\{S_{10}} = (f(0) + 2f(0.3) + 2f(0.6) + \cdots + 2f(1.2) + f(2))\frac{{\Delta x}}{3}\\ \approx 0.642551\end{aligned}\)