Question

Determine the area of the region bounded by x = y²-y-6 and x = 2y +4.

Answer 1

$$y^2-y-6=2y+4$$

$$y^2-3y-10=0$$

$$(y+2)(y-5)=0$$

$$y_1=-2, y_2=5$$

$$A=\displaystyle\int_{-2}^{5}(2y+4-(y^2-y-6)dy$$

$$=[-\dfrac{y^3}{3}+\dfrac{3y^2}{2}+10y]\begin{matrix} 5 \\ -2 \end{matrix}$$

$$=-\dfrac{125}{3}+\dfrac{75}{2}+50-(\dfrac{8}{3}+6-20)=\dfrac{343}{6}({units}^2)$$