Question

решить хотя бы несколько задач.

Answer 1

$a)\ \ y=-\dfrac{1}{7}\, x^2\ ,\ \ y=0\ ,\ \ x=-7\\\\\displaystyle S=-\int\limits_{-7}^0\, (-\dfrac{1}{7}\, x^2)\, dx=\frac{1}{7}\cdot \frac{x^3}{3}\Big|_{-7}^0=0-\frac{-7^3}{7\cdot 3}=\frac{7^2}{3}=\frac{49}{3}\\\\\\b)\ \ y=-x^2+4\ ,\ \ y=0\\\\S=\int\limits_{-2}^2\, (-x^2+4)\, dx=2\int\limits^2_0\, (-x^2+4)\, dx=2\cdot \Big(-\frac{x^3}{3}+4x\Big)\Big|_0^2=\\\\\\=2\cdot \Big(-\frac{8}{3}+8\Big)=2\cdot \frac{16}{3}=\frac{32}{3}\\\\\\c)\ \ y=\sqrt{x+9}\ ,\ x=0\ ,\ y=0$

$S=\int\limits_{-9}^0\, \sqrt{x+9}\ dx=\dfrac{2\sqrt{(x+9)^3}}{3}\Big|_{-9}^0=\dfrac{2}{3}\cdot \Big(\sqrt{9^3}-0\Big)=\dfrac{2}{3}\cdot 3^3=2\cdot 3^2=18$