Find the slope of the line through the origin that splits the region bounded by the parabola y = x-x^2 and the x-axis into two regions of equal area

Answer 1

y=x-x^2 is a parabola

what is the slope of the line through the origin that splits the area bounded by that parabola and the x axis

the area = integral of (x-x^2) = x^2 -x^3/3 evaluated from x=0 to x=1

which = 1-1/3 = 2/3 square units

you want a line that splits that into 2 equal areas. each will have area = (2/3)/2 = 1/3

let the intersection of the line and the parabola be the point (b,h)= (b, b-b^2)

then the area of the lower half will =

the area of a triangle = one half base times height = bh/2 = b(b-b^2)/2 = (b^2-b^3)/2

plus the area of the integral of (x-x^2) = x^2 -x^3/3 evaluated from x=b to x=1 which = 2/3 - b^2+b^3/3

1/3 = (b^2-b^3)/2 + 2/3 - b^2 +b^3/3

2/3 -1/3 = b^2 -b^3/2 - b^2/2 +b^3/3

1/3 = b^2/2 +b^3/6

3b^2 +b^3 = 2

there are three solutions, you want one of them

where b should be between 1/2 and 1, cubic equations are not easy to solve

the parabola x-x^2 in vertex form is

y=-(x^2 - x + 1/4) +1/4

= -(x-1/2)^2 + 1/4 with vertex = (1/2, 1/4) = maximum point

b=about .73205

then h= b-b^2 = .73205 -(.73205)^2

h/b = (b-b^2)/b = 1-b

slope of the area splitting line = 1-.73205 = .26795

slope = about .26795

that's a calculus solution

a little tedious, so no guarantees the above is error free, but

the b value appears to be in the right range of value about midway between 1/2 and 1

a rough check on the solution is use a graphing calculator and graph the parabola and the line y=.27x

and see if the two areas look about equal

just a rough visual guess is that you want a line with a little smaller slope such as y=.22x but that depends on how good my eyesight is

another check is evaluate the integral from x=0 to x=.73

for the integral giving you the top half of the split area

= integral of (x-x^2)- .27x = x^2 -x^3/3 -.27x^2/2 = .87x^2 -x^3/3

evaluate from 0 to .73

= .87(.73)^2 - (.73)^3/3= about .334 = about 1/3

which checks out surprisingly well

go with slope = .27 or .26795 if you want more decimal places