1) y=x²+1 парабола, ветви вверх, вершина в точке (0,1)
y=3-x² парабола, ветви вниз, вершина в точке (0,3)
Точки пересечения парабол: х²+1=3-х² , 2х²=2 , х²=1 , х=1 и х= -1 .
![S=\int\limits^1_{-1}\, (3-x^2-(x^2+1))\, dx=\int\limits^1_{-1}\, (2-2x^2)\, dx=(2x-\frac{2}{3}x^3)\Big |_{-1}^1=\\\\=2-\frac{2}{3}-(-2+\frac{2}{3})=4-\frac{4}{3}=\frac{8}{3}](https://tex.z-dn.net/?f=S%3D%5Cint%5Climits%5E1_%7B-1%7D%5C%2C+%283-x%5E2-%28x%5E2%2B1%29%29%5C%2C+dx%3D%5Cint%5Climits%5E1_%7B-1%7D%5C%2C+%282-2x%5E2%29%5C%2C+dx%3D%282x-%5Cfrac%7B2%7D%7B3%7Dx%5E3%29%5CBig+%7C_%7B-1%7D%5E1%3D%5C%5C%5C%5C%3D2-%5Cfrac%7B2%7D%7B3%7D-%28-2%2B%5Cfrac%7B2%7D%7B3%7D%29%3D4-%5Cfrac%7B4%7D%7B3%7D%3D%5Cfrac%7B8%7D%7B3%7D)
2) y=√x верхняя ветвь параболы у²=х , ветви вправо, вершина в точке (0,0) .
у=√(4-3х) верхняя ветвь параболы у²=4-3х , ветви влево, вершина в точке (0,4/3) .
Точки пересечения парабол: х=4-3х , 4х=4 , х=1 и х= -1 .
![S=\int\limits^1_0\, \sqrt{x}\, dx+\int\limits_1^{\frac{4}{3}}\, \sqrt{4-3x}\, dx=\frac{x^{\frac{3}{2}}}{3/2}\Big |_0^1+\frac{-1}{3}\cdot \frac{(4-3x)^{\frac{3}{2}}}{3/2}\Big |_1^{\frac{4}{3}}=\\\\=\frac{2}{3}\cdot (1-0)-2\cdot (0-1)=\frac{2}{3}+2=2\frac{2}{3}](https://tex.z-dn.net/?f=S%3D%5Cint%5Climits%5E1_0%5C%2C+%5Csqrt%7Bx%7D%5C%2C+dx%2B%5Cint%5Climits_1%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%5C%2C+%5Csqrt%7B4-3x%7D%5C%2C+dx%3D%5Cfrac%7Bx%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%7B3%2F2%7D%5CBig+%7C_0%5E1%2B%5Cfrac%7B-1%7D%7B3%7D%5Ccdot+%5Cfrac%7B%284-3x%29%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%7B3%2F2%7D%5CBig+%7C_1%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%3D%5C%5C%5C%5C%3D%5Cfrac%7B2%7D%7B3%7D%5Ccdot+%281-0%29-2%5Ccdot+%280-1%29%3D%5Cfrac%7B2%7D%7B3%7D%2B2%3D2%5Cfrac%7B2%7D%7B3%7D)