Find the values of c such that the area of the region bounded by the parabolas
y = 16x2 − c2 and y = c2 − 16x2
is 144

Question

Find the values of c such that the area of the region bounded by the parabolas 
y = 16x2 − c2 and y = c2 − 16x2
 is 144

psymon · Answer

Well, these functions are completely opposite. One is the upside down version of the other. What this also means is that each function contributes half of the toal area. So if you can find a value of c such that integrating ONE of the above functions gives an area of 72, then you have your answer.

anonymous · Answer

Okay. So.. you set the two equal to each other to find the bounds, right ?

psymon · Answer

Right.

anonymous · Answer

so you would get x=plus or minus $$\sqrt{(1/16)c ^{2}}$$

anonymous · Answer

Yes ?

psymon · Answer

You put it in a different way than I would of xD 
$$x=\pm \frac{ c }{ 4 } $$

anonymous · Answer

Oh. Yours is prettier, lol. But are they the same thing, how does one simplify to the other ? 0.0

psymon · Answer

Well, when you did the square root you forgot the plus/minus xD But when we factored we got 2(16x^2 - c^2), which is a difference of squares that I can factor to 2(4x+c)(4x-c). From here, I just set each factor = 0
4x + c = 0
4x - c = 0

anonymous · Answer

Ooooh, okay. Gotcha, cool. So, once you get that, you would set the integration of the subtraction of the two formulas equal to 144 right ?

psymon · Answer

Well, we don't really want to do that, it'll make our work more complex. What we have our opposite function, one as a parabola that opens down and one as a parabola that opens up. Because of the symmetry, each function will contribute to half of the area, so 72. So all we really need to do is deal with one function

|dw:1377480896968:dw|