Решебник по алгебре 11 класс Никольский Параграф 6. Первообразная и интеграл Задание 70

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\[\textbf{а)}\ y = x^{2} - \pi x;\]

\[y = \sin x;\]

\[\sin x = x^{2} - \pi x\]

\[x_{1} = 0;\ \ x_{2} = \pi;\]

\[y(0) = 0;\]

\[y(\pi) = 0.\]

\[x \in (0;\ \pi):\]

\[S = \int_{0}^{\pi}{\left( \sin x - \left( x^{2} - \pi x \right) \right)\text{dx}} =\]

\[= \int_{0}^{\pi}{\left( \sin x - x^{2} + \pi x \right)\text{dx}} =\]

\[= \left. \ - \cos x - \frac{x^{3}}{3} + \frac{x\pi^{2}}{2} \right|_{0}^{\pi} =\]

\[= 1 + \frac{\pi^{3}}{6} + 1 = \frac{\pi^{3}}{6} + 2\ (кв.\ ед.).\]

\[Ответ:\ \frac{\pi^{3}}{6} + 2\ (кв.\ ед.).\]

\[\textbf{б)}\ y = \sin x;\ y = \cos x;\]

\[x = \frac{\pi}{4};\ \ x = \frac{5\pi}{4};\]

\[S = \int_{\frac{\pi}{4}}^{\frac{5\pi}{4}}{\sin x\text{dx}} - \int_{\frac{\pi}{4}}^{\frac{5\pi}{4}}{\cos x\text{dx}} =\]

\[= \int_{\frac{\pi}{4}}^{\frac{5\pi}{4}}{\left( \sin x - \cos x \right)\text{dx\ }} =\]

\[= \left. \ - \cos x - \sin x \right|_{\frac{\pi}{4}}^{\frac{5\pi}{4}} =\]

\[= \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} =\]

\[= 2\sqrt{2}\ (кв.\ ед.).\]

\[Ответ:2\sqrt{2}\ кв.\ ед.\]