The area (in square units) of the region bounded by the parabola y^2=4(x-2) and the line y…

Question

The area (in square units) of the region bounded by the parabola $y^2=4(x-2)$ and the line $y=2x-8$ is:

Accepted Answer

Write $x=\dfrac{y^2}{4}+2$ (parabola) and $x=\dfrac{y+8}{2}$ (line). Setting equal:

$$y^2+8=2(y+8) \;\Rightarrow\; y^2-2y-8=0 \;\Rightarrow\; y=4,-2.$$

$$\text{Area}=\int_{-2}^{4}\left[\frac{y+8}{2}-\left(\frac{y^2}{4}+2\right)\right]dy=\left[\frac{y^2}{4}+2y-\frac{y^3}{12}\right]_{-2}^{4}=\frac{20}{3}-\left(-\frac73\right)=9.$$

Solutions · 1