areas between curves

**zaddie21** · March 14th 2009, 12:46 PM

there is a line through the origin that divides the region bounded by the parabola y= 8x-2x^2 and the x-axis into two regions of equal area. What is the slope of the line?

I can't figure it out. I don't know how to start

**skeeter** · March 14th 2009, 12:57 PM

Originally Posted by zaddie21

there is a line through the origin that divides the region bounded by the parabola y= 8x-2x^2 and the x-axis into two regions of equal area. What is the slope of the line?

I can't figure it out. I don't know how to start

calculate the area in quad I bounded by the parabola and the x-axis, call that area $A$ .

let the line be $y = mx$

find the intersection of the line and parabola in quad I ...

$mx = 8x - 2x^2$

$0 = (8-m)x - 2x^2$

$0 = x(8 - m - 2x)$

x-values of intersection are $x = 0$ and $x = \frac{8-m}{2}$

$\frac{A}{2} = \int_0^{\frac{8-m}{2}} (8x-2x^2) - mx \, dx$

evaluate the integral and solve for $m$ ... have fun with the algebra.

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