Chapter 4 - Integration - 4.6 Exercises - Page 310: 13

$$\eqalign{ & {\text{Trapezoidal Rule}} \approx 0.3415 \cr & {\text{Simpson's Rule}} \approx 0.3720 \cr & {\text{Graphing utility}} \approx 0.393 \cr} $$

$$\eqalign{ & \int_0^1 {\sqrt x \sqrt {1 - 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{{1 - 0}}{4} = \frac{1}{4},{\text{ then,}} \cr & {x_0} = 0,{\text{ }}{x_1} = \frac{1}{4},{\text{ }}{x_2}{\text{ = }}\frac{1}{2}{\text{, }}{x_3} = \frac{3}{4},{\text{ }}{x_4} = 1 \cr & \int_0^1 {\sqrt x \sqrt {1 - x} } dx \cr & \approx \frac{1}{8}\left[ {f\left( 0 \right) + 2f\left( {\frac{1}{4}} \right) + 2f\left( {\frac{1}{2}} \right) + 2f\left( {\frac{3}{4}} \right) + f\left( 1 \right)} \right] \cr & \approx \frac{1}{8}\left[ {0 + 2\sqrt {\frac{1}{4}} \sqrt {1 - \frac{1}{4}} + 2\sqrt {\frac{1}{2}} \sqrt {1 - \frac{1}{2}} + 2\sqrt {\frac{3}{4}} \sqrt {1 - \frac{3}{4}} + 2\sqrt 1 \sqrt 0 } \right] \cr & \approx \frac{1}{8}\left[ {0 + \frac{{\sqrt 3 }}{2} + 1 + \frac{{\sqrt 3 }}{2} + 0} \right] \cr & {\text{Simplifying}} \cr & \int_0^1 {\sqrt x \sqrt {1 - x} } dx \approx 0.3415 \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_0^1 {\sqrt x \sqrt {1 - x} } dx \approx \cr & \approx \frac{{1 - 0}}{{3\left( 4 \right)}}\left[ {0 + 4\sqrt {\frac{1}{4}} \sqrt {1 - \frac{1}{4}} + 2\sqrt {\frac{1}{2}} \sqrt {1 - \frac{1}{2}} + 4\sqrt {\frac{3}{4}} \sqrt {1 - \frac{3}{4}} + 0} \right] \cr & {\text{Simplifying}} \cr & \int_0^1 {\sqrt x \sqrt {1 - x} } dx \approx 0.3720 \cr & \cr & {\text{Using a graphing utility we obtain}} \cr & \int_0^1 {\sqrt x \sqrt {1 - x} } dx \approx 0.393 \cr} $$