Integration/Solving A Constant To Divide An Area

[Lancelot59]

With this particular problem you are given the functions y=x^{2}, y=4, and y=b.

The object is to solve for a value of b that will split the area between y=4 and y=x^{2} in half.
[PLAIN]http://www4c.wolframalpha.com/Calculate/MSP/MSP263419aghg2f97i2944e0000580a2hgi735b957c?MSPStoreType=image/gif&s=42&w=399&h=201

The approach I took was to include y=b in the integral, and then subtract the top half from the bottom half and equate that to zero.

\int_{-2}^{2} {(4-x^2-b)}\ = \int_{-2}^{2} {(b-x^2)}

All I did then was solve the integrals, and isolate b to get an answer of 4. However the answer is 4^2/3. I'm also stuck on a similar problem where I need to solve for a constant where two functions have to enclose a specific area.

 

[gabbagabbahey]

Your integral for the top half is incorrect. Using the "vertical slices" method, the lower limit on y is y=b when -\sqrt{b}<x<\sqrt{b}, and y=x^2 when |x|\geq\sqrt{b}. So, you might want to either use horizontal slices for the top half, or not bother computing that integral at all and instead compute the area between y=x^2 and y=4 and set that equal to twice your bottom half.

 

[Lancelot59]

Horizontal slices sounds like a better idea. I did come up with the +-root b being the limits, but I decided to go with negative two and two instead...apparently that didn't work. I guess just integrating for one half only works out nicely. Then I can just go from zero to 4.

 

[Lancelot59]

Okay, so I did it with respect to Y instead:

<br /> \int_{0}^{b} {\sqrt{y}}\ = \int_{0}^{4} {\sqrt{y}}\ - \int_{0}^{b} {\sqrt{y}}<br />

then got here:

<br /> [\sqrt{b} - \sqrt{0}] = [\sqrt{4} - \sqrt{0}] - [\sqrt{b} - \sqrt{0}]<br />

then:
<br /> 2\sqrt{b}= \sqrt{4}<br />

<br /> 2\sqrt{b}= 2<br />

<br /> b= 1<br />

Things aren't going very well.

 

[happyg1]

I found the desired area to be 8/3 (I just chopped it in half and did 0 to 2). Then integrated \frac{8}{3}=\int_{0}^{\sqrt b} <br /> b-x^2 dx
Solved for b...

 

[Lancelot59]

You mean
\int_{0}^{\sqrt b}<br /> { [b-x^2]} dx <br />

? I'll give it a try.

 

[happyg1]

Yes, My tex skills are a little rusty. i edited it and fixed it..

 

[Lancelot59]

Mine are horrible. I just figured it out within the last hour to ask this question. :p

Also, that doesn't work. The answer is supposed to be the cube root of 16...I am confused.

 

[happyg1]

set it equal to 8/3! It works!

 

[Lancelot59]

And the 8/3 is half the parabola from 0 to 2? Attempt # 4ish...

 

[Lancelot59]

Well, here's what I did:

<br /> \frac{8}{3}=\int_{0}^{\sqrt b} <br /> b-x^2 dx<br />

<br /> \frac{8}{3}=[b-{\sqrt b}^2] - [b-{\sqrt 0}^2]<br />

<br /> \frac{8}{3}= b-b-b<br />

<br /> \frac{8}{3}= -b<br />

I'm really confused now.

 

[happyg1]

<br /> <br /> \frac{8}{3}=\int_{0}^{\sqrt b} <br /> b-x^2 dx<br /> <br /> =bx-\frac{1}{3}x^3 from 0 to \sqrt b

 

[Cyosis]

I suggest you review basic integration. You're just plugging the numbers into the integrand instead of integrating.

 

[Lancelot59]

...Fail. I've been doing that a lot recently. I don't know why.