How do you find the area of the region enclosed by the line y=2x+4 and the parabola y=4-x^2 in the xy-plane? Thank you
The line and the parabola intersect at the points where the x-coordinate is 0 and -2. (Obtained by solving the 2 equations.)
$\displaystyle Area=\int\limits_{-2}^{0}(4-x^2)-(2x+4) dx$
$\displaystyle =\int\limits_{-2}^{0} (-x^2 - 2x)dx$
$\displaystyle =-\frac{x^3}{3}-x^2 | $ (from -2 to 0)
$\displaystyle =0-(\frac{8}{3}-4)$
$\displaystyle =\frac{4}{3}$