Chapter 4 - Integration - 4.6 Exercises - Page 310: 18

$$\eqalign{ & {\text{Trapezoidal Rule}} \approx 1.9100 \cr & {\text{Simpson's Rule}} \approx 1.91014 \cr & {\text{Graphing utility}} \approx 1.910098 \cr} $$

$$\eqalign{ & \int_0^{\pi /2} {\sqrt {1 + {{\sin }^2}x} } 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{{\pi /2 - 0}}{4} = \frac{\pi }{8},{\text{ then,}} \cr & {x_0} = 0,{\text{ }}{x_1} = \frac{\pi }{8},{\text{ }}{x_2} = \frac{\pi }{4}{\text{, }}{x_3} = \frac{{3\pi }}{8},{\text{ }}{x_4} = \frac{\pi }{2} \cr & f\left( {{x_0}} \right) = f\left( 0 \right) = \sqrt {1 + {{\sin }^2}\left( 0 \right)} = 1 \cr & f\left( {{x_1}} \right) = f\left( {\frac{\pi }{8}} \right) = \sqrt {1 + {{\sin }^2}\left( {\frac{\pi }{8}} \right)} \cr & f\left( {{x_2}} \right) = f\left( {\frac{\pi }{4}} \right) = \sqrt {1 + {{\sin }^2}\left( {\frac{\pi }{4}} \right)} = \sqrt {\frac{3}{2}} \cr & f\left( {{x_3}} \right) = f\left( {\frac{{3\pi }}{8}} \right) = \sqrt {1 + {{\sin }^2}\left( {\frac{{3\pi }}{8}} \right)} \cr & f\left( {{x_4}} \right) = f\left( {\frac{\pi }{2}} \right) = \sqrt {1 + {{\sin }^2}\left( {\frac{\pi }{2}} \right)} = \sqrt 2 \cr & {\text{Therefore,}} \cr & \int_0^{\pi /2} {\sqrt {1 + {{\sin }^2}x} } dx \approx \frac{\pi }{{2\left( 8 \right)}}\left[ {1 + 2\sqrt {1 + {{\sin }^2}\left( {\frac{\pi }{8}} \right)} + 2\sqrt {\frac{3}{2}} } \right] \cr & + \frac{\pi }{{2\left( 8 \right)}}\left[ {2\sqrt {1 + {{\sin }^2}\left( {\frac{{3\pi }}{8}} \right)} + \sqrt 2 } \right] \cr & {\text{Simplifying by using a calculator}} \cr & \int_0^{\pi /2} {\sqrt {1 + {{\sin }^2}x} } dx \approx 1.9100 \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^{\pi /2} {\sqrt {1 + {{\sin }^2}x} } dx \approx \frac{\pi }{{3\left( 8 \right)}}\left[ {1 + 4\sqrt {1 + {{\sin }^2}\left( {\frac{\pi }{8}} \right)} + 2\sqrt {\frac{3}{2}} } \right] \cr & + \frac{\pi }{{3\left( 8 \right)}}\left[ {4\sqrt {1 + {{\sin }^2}\left( {\frac{{3\pi }}{8}} \right)} + \sqrt 2 } \right] \cr & {\text{Simplifying by using a calculator}} \cr & \int_3^{3.1} {\cos {x^2}} dx \approx 1.91014 \cr & \cr & {\text{Using a graphing utility we obtain}} \cr & \int_0^{\pi /2} {\sqrt {1 + {{\sin }^2}x} } dx \approx 1.910098 \cr} $$