Решебник по алгебре 11 класс Никольский Параграф 6. Первообразная и интеграл Задание 57

\[\boxed{\mathbf{57}.}\]

\[\textbf{а)}\ y = x^{3};\ \ x = 1;\ \ y = 0;\]

\[F(x) = \frac{x^{4}}{4};\]

\[S = \int_{0}^{1}{x^{3}\text{dx}} = F(1) - F(0) =\]

\[= \frac{1}{4} - 0 = \frac{1}{4}\ кв.\ ед.\]

\[\textbf{б)}\ x^{3};\ \ x = - 1;\ \ y = 0;\]

\[F(x) = \frac{x^{4}}{4};\]

\[S = - \int_{- 1}^{0}{x^{3}\text{dx}} =\]

\[= - \left( F(0) - F( - 1) \right) = \frac{1}{4} - 0 =\]

\[= \frac{1}{4}\ кв.\ ед.\]

\[\textbf{в)}\ x^{3};\ \ x = - 1;\ \ x = 1;\ y = 0;\]

\[F(x) = \frac{x^{4}}{4};\]

\[S_{1} = \int_{0}^{1}{x^{3}\text{dx}} = \frac{1}{4};\]

\[S_{2} = - \int_{- 1}^{0}{x^{3}\text{dx}} = \frac{1}{4};\]

\[S = S_{1} + S_{2} = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}\ кв.\ ед.\]