Solve Parabola for Equal Area: y=a(x^2) in Calc II

[RUStudent]

Homework Statement

Find the parabola y=a(x^2) that divides the area under the curve y=x(1-x) over [0,1] into two regions of equal area.

Homework Equations

I set the two equations equal to each other to solve that the intersection point is x=1/(1+a).
I solved for the entire area "definite integral x(1-x) from [0,1] dx" = 1/6.

The Attempt at a Solution

I attempt half the area with the two definite integrals "a(x^2) from [0,(1/(1+a))]" + the integral "x(1-x) from [(1/(1+a)), 1]" set equal to 1/12 (half the area). But I can not solve for a.

It looks like there is Linear algebra but I am only in a Calc II class so it should not be too hard. Please help.

 

[Dick]

The area under y=x(1-x) is supposed to split into two regions of area 1/12. One of them is the region bounded by y=x(1-x) above and y=ax^2 below between 0 and 1/(1+a). Figure out what that is and set it equal to 1/12. The other region will take care of itself.

 

[RUStudent]

The area under y=x(1-x) is supposed to split into two regions of area 1/12. One of them is the region bounded by y=x(1-x) above and y=ax^2 below between 0 and 1/(1+a). Figure out what that is and set it equal to 1/12. The other region will take care of itself.

That is what I have done. But I come up with an equation too hard to solve for a. I believe my problem lies in my algebra.

 

[Dick]

That is what I have done. But I come up with an equation too hard to solve for a. I believe my problem lies in my algebra.

Ok. So what do you get for the integral in terms of a? I get an equation that looks like it might be a cubic, but when you do the algebra the cubic term cancels out and it becomes a quadratic.

 

[RUStudent]

That is what I thought I was doing wrong but I have an addition of the cubics. My integral is x-x^2 from 0 to (1/1+a) - ax^2 from x to (1/1+a).

 

[Dick]

Right so far except the second limit is also 0 to 1/(1+a), just a typo I hope. So put the limits in and subtract. Bring everything to a common denominator and set it equal to 1/6. The cubic terms canceled for me. If you show us the intermediate steps maybe we can see what went wrong.

 

[RUStudent]

Yes sorry that was a typo. After integration I get (x^2/x)-(x^3/3)-a(x^3/3). I set this equal to 1/12 but as you see the cubic terms do not cancel.

 

[Dick]

Put x=1/(1+a) and THEN try and do the algebra. The cubic parts in a will cancel.

 

[RUStudent]

I'm not sure I understand. After I integrate substitute x for (1/1+a). If I do this I get "((1/1+a)^2/2)-((1/1+a)^3/3)-a((1/1+a)^3/3) and they still do not cancel.

 

[Dick]

You haven't done any simplification yet. I told you, bring them to a common denominator and sum them all into one term.

 

[RUStudent]

I'm confused. Now I have (1/1+a)^2(3-2(1/1+a)-2a(1/1+a))=1/2. I still have the cubic terms.

 

[Dick]

You aren't done yet! Add up the terms in the parentheses!

 

[RUStudent]

Yes I did that but I don't get it, there is still a cubic.

 

[Dick]

Yes I did that but I don't get it, there is still a cubic.

Argh. What is 3-2/(1+a)-2a/(1+a)? Common denominator. Add them.

 

[RUStudent]

Common denominator is (1+a) so I have ((3(1+a)-2-2a)/(1+a))(1/(1+a))^2. This would give me (3(1+a)-2-2a)(1/(1+a))^3=1/2. I am so frustrated. I don't get it!

 

[Dick]

You are doing everything right but you keep taking one step and then stopping. 3(1+a)-2-2a=3+3a-2a-2=1+a. Do you see the cancellation now??

 

[RUStudent]

Wow, thank you so much, I don't know why I couldn't see that. a=0.41421. Thanks again.