Sketch the curves y = x^2 and y + 2x = 8. Find the area of the region bound by the x-axis and these two curves by Writing the area as a single denite integral with respect to y. I honestly have no idea how to do this..

Question

Sketch the curves y = x^2 and y + 2x = 8. Find the area of the region bound by the x-axis and these two curves by Writing the area as a single denite integral with respect to y. I honestly have no idea how to do this..

nenadmatematika · Answer

first, find two intersection points of parabola x^2 and line y=-2x+8, and those points will be the boundaries of your integral, so you have integral of (line minus parabola) in those boundaries and that 's how you'll get area

nikvist · Answer

$$y=x^2\quad,\quad y+2x=8$$$$x^2+2x-8=(x-2)(x+4)=0$$$$x=2\quad\Rightarrow\quad y=4$$$$S=\int\limits_0^4\left(4-\frac{y}{2}-\sqrt{y}\right)dy=\left(4y-\frac{y^2}{4}-\frac{2}{3}y^{3/2}\right)_0^4=$$$$=16-4-\frac{16}{3}=\frac{20}{3}$$