Решебник по алгебре 11 класс Никольский Параграф 6. Первообразная и интеграл Задание 67

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\[\textbf{а)}\ y = \frac{x^{2}}{2};\]

\[x = 1;x = 3;\ \ y = 0;\]

\[S = \int_{1}^{3}{\frac{x^{2}}{2}\text{dx}} = \left. \ \frac{x^{3}}{3} \right|_{1}^{3} =\]

\[= \frac{3^{3}}{6} - \frac{1^{3}}{6} = \frac{27}{6} - \frac{1}{6} = \frac{26}{6} =\]

\[= 4\frac{1}{3}\ кв.\ ед.\]

\[Ответ:4\frac{1}{3}\ кв.\ ед.\]

\[\textbf{б)}\ y = \sqrt{2x};\]

\[x = 1;\ \ y = 0;\]

\[S = \int_{0}^{1}{\sqrt{2x}\text{dx}} = \sqrt{2}\int_{0}^{1}{x^{\frac{1}{2}}\text{dx}} =\]

\[= \left. \ \frac{2\sqrt{2}x}{3} \right|_{0}^{1} = \frac{2\sqrt{2}}{3}\left( 1^{\frac{3}{2}} - 0^{\frac{3}{2}} \right) =\]

\[= \frac{2\sqrt{2}}{3}\ кв.\ ед.\]

\[Ответ:\frac{2\sqrt{2}}{3}\ кв.\ ед.\]