Chapter 4 - Integration - 4.6 Exercises - Page 310: 15

$$\eqalign{ & {\text{Trapezoidal Rule}} \approx 0.550 \cr & {\text{Simpson's Rule}} \approx 0.548 \cr & {\text{Graphing utility}} \approx 0.5492 \cr} $$

$$\eqalign{ & \int_0^{\sqrt {\pi /2} } {\sin {x^2}} dx \cr & {\text{*Using the trapezoidal Rule }}\left( {{\text{THEOREM 4}}{\text{.17}}} \right) \cr & \int_a^b {f\left( x \right)} dx \approx \frac{{b - a}}{{2n}}\left[ {f\left( {{x_0}} \right) + 2f\left( {{x_1}} \right) + \cdots 2f\left( {{x_{n - 1}}} \right) + f\left( {{x_n}} \right)} \right] \cr & {\text{For }}n = 4,{\text{ }}\Delta x = \frac{{b - a}}{n} = \frac{{\sqrt {\pi /2} - 0}}{4} = \frac{{\sqrt {\pi /2} }}{4},{\text{ then,}} \cr & {x_0} = 0,{\text{ }}{x_1} = \frac{{\sqrt {\pi /2} }}{4},{\text{ }}{x_2}{\text{ = }}\frac{{\sqrt {\pi /2} }}{2}{\text{, }}{x_3} = \frac{{3\sqrt {\pi /2} }}{4},{\text{ }}{x_4} = \sqrt {\pi /2} \cr & f\left( {{x_0}} \right) = f\left( 0 \right) = \sin {\left( 0 \right)^2} = 0 \cr & f\left( {{x_1}} \right) = f\left( {\frac{{\sqrt {\pi /2} }}{4}} \right) = \sin {\left( {\frac{{\sqrt {\pi /2} }}{4}} \right)^2} \cr & f\left( {{x_2}} \right) = f\left( {\frac{{\sqrt {\pi /2} }}{2}} \right) = \sin {\left( {\frac{{\sqrt {\pi /2} }}{2}} \right)^2} \cr & f\left( {{x_3}} \right) = f\left( {\frac{{3\sqrt {\pi /2} }}{4}} \right) = \sin {\left( {\frac{{3\sqrt {\pi /2} }}{4}} \right)^2} \cr & f\left( {{x_4}} \right) = f\left( {\sqrt {\pi /2} } \right) = \sin {\left( {\sqrt {\pi /2} } \right)^2} \cr & \int_0^{\sqrt {\pi /2} } {\sin {x^2}} dx \approx \frac{{\sqrt {\pi /2} }}{8}\left[ {0 + 2\sin {{\left( {\frac{{\sqrt {\pi /2} }}{4}} \right)}^2}} \right] \cr & + \frac{{\sqrt {\pi /2} }}{8}\left[ { + 2\sin {{\left( {\frac{{\sqrt {\pi /2} }}{2}} \right)}^2} + 2\sin {{\left( {\frac{{3\sqrt {\pi /2} }}{4}} \right)}^2} + \sin {{\left( {\sqrt {\frac{\pi }{2}} } \right)}^2}} \right] \cr & {\text{Simplifying by using a calculator}} \cr & \int_0^{\sqrt {\pi /2} } {\sin {x^2}} dx \approx 0.550 \cr & \cr & {\text{*Using the Simpson's Rule }}\left( {{\text{THEOREM 4}}{\text{.19}}} \right) \cr & \int_a^b {f\left( x \right)} dx \approx \frac{{b - a}}{{3n}}\left[ {f\left( {{x_0}} \right) + 4f\left( {{x_1}} \right) + 2f\left( {{x_2}} \right) + 4f\left( {{x_3}} \right) + \cdots } \right. \cr & \left. { + 4f\left( {{x_{n - 1}}} \right) + f\left( {{x_n}} \right)} \right] \cr & \int_0^{\sqrt {\pi /2} } {\sin {x^2}} dx \approx \frac{{\sqrt {\pi /2} }}{{12}}\left[ {0 + 4\sin {{\left( {\frac{{\sqrt {\pi /2} }}{4}} \right)}^2}} \right] \cr & + \frac{{\sqrt {\pi /2} }}{{12}}\left[ { + 2\sin {{\left( {\frac{{\sqrt {\pi /2} }}{2}} \right)}^2} + 4\sin {{\left( {\frac{{3\sqrt {\pi /2} }}{4}} \right)}^2} + \sin {{\left( {\sqrt {\frac{\pi }{2}} } \right)}^2}} \right] \cr & {\text{Simplifying by using a calculator}} \cr & \int_0^{\sqrt {\pi /2} } {\sin {x^2}} dx \approx 0.548 \cr & \cr & {\text{Using a graphing utility we obtain}} \cr & \int_0^{\sqrt {\pi /2} } {\sin {x^2}} dx \approx 0.5492 \cr} $$