Решебник по алгебре 11 класс Никольский Параграф 6. Первообразная и интеграл Задание 53

\[\boxed{\mathbf{53}.}\]

\[\textbf{а)}\ y = x^{2};\ \ x = 3;\ \ x = 5;\ \ y = 0;\]

\[S = \int_{3}^{5}{x^{2}\text{dx}} = \left. \ \frac{x^{3}}{3} \right|_{3}^{5} = \frac{5^{3}}{3} - \frac{3^{3}}{3} =\]

\[= \frac{125}{3} - \frac{27}{3} = \frac{98}{3} = 32\frac{2}{3}\ кв.\ ед.\]

\[\textbf{б)}\ y = x^{3};\ x = 1;\ \ x = 2;\ \ y = 0;\]

\[S = \int_{1}^{2}{x^{3}\text{dx}} = \left. \ \frac{x^{4}}{4} \right|_{1}^{2} = \frac{2^{4}}{4} - \frac{1}{4} =\]

\[= 4 - 0,25 = 3,75\ кв.\ ед.\]

\[\textbf{в)}\ y = \frac{1}{x};\ \ x = 1;\ \ x = 4;\ \ y = 0;\]

\[S = \int_{1}^{4}{\frac{1}{x}\text{dx}} = \left. \ \ln|x| \right|_{1}^{4} =\]

\[= \ln 4 - \ln 1 = \ln 4\ кв.\ ед.\]