anonymous
  • anonymous
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..
Mathematics
chestercat
  • chestercat
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nenadmatematika
  • nenadmatematika
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
  • nikvist
\[y=x^2\quad,\quad y+2x=8\]\[x^2+2x-8=(x-2)(x+4)=0\]\[x=2\quad\Rightarrow\quad y=4\]\[S=\int\limits_0^4\left(4-\frac{y}{2}-\sqrt{y}\right)dy=\left(4y-\frac{y^2}{4}-\frac{2}{3}y^{3/2}\right)_0^4=\]\[=16-4-\frac{16}{3}=\frac{20}{3}\]
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