Question

# Find the region enclosed by the curves x = 3y^2 and x = y^2 + 7 List the points of intersection of these curves from bottom to top in the form (x, y).

Analyzing functions
Find the region enclosed by the curves $$\displaystyle{x}={3}{y}^{{2}}{\quad\text{and}\quad}{x}={y}^{{2}}+{7}$$
List the points of intersection of these curves from bottom to top in the form (x, y).

2021-01-24
$$\displaystyle{x}={3}{y}^{{2}},{x}={y}^{{2}}+{7}$$
$$\displaystyle{3}{y}^{{2}}={y}^{{2}}+{7}$$
$$\displaystyle{2}{y}^{{2}}={7}$$
$$\displaystyle{y}^{{2}}=\frac{{7}}{{2}}$$
$$\displaystyle{y}=\pm\sqrt{{\frac{{7}}{{2}}}}$$
$$\displaystyle{y}=\sqrt{{\frac{{7}}{{2}}}}$$
$$\displaystyle{x}={3}\times\frac{{7}}{{2}}=\frac{{21}}{{2}}$$
or $$\displaystyle{y}=-\sqrt{{\frac{{7}}{{2}}}}$$
$$\displaystyle{x}=\frac{{21}}{{2}}$$
The points: $$\displaystyle{\left(\frac{{21}}{{2}},\sqrt{{\frac{{7}}{{2}}}}\right)},{\left(\frac{{21}}{{2}},-\sqrt{{\frac{{7}}{{2}}}}\right)}$$.