Area between curves

[InterserveVB]

How would I go about these?

Find the number b such that the line y = b divides the region bounded by the curves y = 7x^2 and y = 5 into two regions with equal areas?

I know the answer is 5 / 2^3/2. I have no clue how to get it or set it up.

Also, Find the positive value of c such that the area of the region bounded by the parabolas y = x^2 - c^2 and y = c^2 - x^2 is 576. How would I do this?

 

[soroban]

Hello, InterserveVB!

Here's the second one . . .
\(\displaystyle \;\;\)Did you make a sketch?

Find the positive value of \(\displaystyle c\) such that the area of the region bounded
by the parabolas \(\displaystyle y\:=\:x^2\,-\,c^2\) and \(\displaystyle y\:=\:c^2\,-\,x^2\) is 576.

\(\displaystyle y\:=\:x^2\,-\,c^2\) is an up-opening parabola with intercepts \(\displaystyle (\pm c,\,0)\,\0,-c^2)\).
\(\displaystyle y \:=\:c^2\,-\,x^2\) is a down-opening parabola with intercepts \(\displaystyle (\pm c,\,0),\0,\,c^2)\).

     *          |          *
               ***
      *    *    |    *    *
       * *      |      * *
    - - * - - - + - - - * - -
       * *      |      * *
      *    *    |    *    *
               ***
     *          |          *

The area between the parabola is: \(\displaystyle \L\:A\;=\;\int^{\;\;\;c}_{-c}\left[(c^2\,-\,x^2)\,-\,(x^2\,-\,c^2)\right]\.dx\)

So we have: \(\displaystyle \L\;A\;= \;\int^{\;\;\;c}_{-c}2(c^2\,-\,x^2)\,dx \;= \;576\)

Integrate, evaluate, and solve the resulting equation for \(\displaystyle c\).

 

[galactus]

You could solve your equations for x in terms of y.

\(\displaystyle 7x^{2}=y\)

\(\displaystyle x=sqrt{\frac{y}{7}}\)

\(\displaystyle 2\int_{0}^{b}\sqrt{\frac{y}{7}}dy=2\int_{b}^{5}\sqrt{\frac{y}{7}}dy\)

Perform the integrations on each side of the equality, set equal to one another and solve for b. I don't believe it's \(\displaystyle \frac{5}{2^{\frac{3}{2}}}\)

I think you have the numerator and denominator in your exponent backwards.

After you find b, check your work by subbing into the above integrals and seeing if you do, indeed, get the same thing. I believe you will.

Since these two integrals represent the area between a line and a parabola divided in two equal parts, integrate the entire region and you should get the same as the above two integrals added together.

\(\displaystyle 2\int_{0}^{5}sqrt{\frac{y}{7}}dy=\frac{20sqrt{35}}{21}\approx5.63...\)

Do each of the integrals above equal approx. 2.82?.