I'm given the function and must work out the x coordinate on f(x), A, which a line dividing the area between the function and the x-axis from [0,2] into 2 parts of equal area passes through. The area total area between the function and x-axis from [0,2] is 4 square units.
I've got it down to
I'm kinda stuck there. Is there another way to do it that doesn't require me to solve polynomial equations?
Let be the point where that line crosses the parabola. Then the line has equation and the area, above that line but below the parabola is given by
Since the area under the entire parabola is 4, find that integral, in terms of , set it equal to 2, and solve for .
If you integrate to get the area under the parabola from x=0 to x=A,
then subtract the area of the triangle, that resulting area is 2.
The area of the triangle is
Alternatively, integrate from x=A to x=2 and add the triangle area.
Those combined areas are also 2.
which leads to