Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 1021

\[\boxed{\mathbf{1021}\mathbf{.}}\]

\[1)\ y = 2 - x^{2}\text{\ \ }и\ \ y = - x\]

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

\[x^{2} - x - 2 = 0\]

\[D = 1^{2} + 4 \bullet 2 = 1 + 8 = 9\]

\[x_{1} = \frac{1 - 3}{2} = - 1\ \ и\ \ \]

\[x_{2} = \frac{1 + 3}{2} = 2.\]

\[S =\]

\[= \int_{- 1}^{2}{\left( \left( 4 - x^{2} \right) - (2 - x) \right)\text{\ dx}} =\]

\[= 2 - \frac{1}{3} - \frac{1}{2} + 4 - \frac{8}{3} + \frac{4}{2} =\]

\[= 6 + 1,5 - 3 = 4,5.\]

\[Ответ:\ \ 4,5.\]

\[2)\ y = 1\ \ и\ \ y = \sin x\ \]

\[при\ 0 \leq x \leq \frac{\pi}{2}\]

\[1 = \sin x\]

\[x = \arcsin 1 + 2\pi n\]

\[x = \frac{\pi}{2} + 2\pi n\ \]

\[x = \frac{\pi}{2}.\]

\[S = \int_{0}^{\frac{\pi}{2}}{\left( 1 - \sin x \right)\text{\ dx}} =\]

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

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

\[= \frac{\pi}{2} + \cos\frac{\pi}{2} - 0 - \cos 0 =\]

\[= \frac{\pi}{2} + 0 - 1 = \frac{\pi}{2} - 1.\]

\[Ответ:\ \ \frac{\pi}{2} - 1.\]