1. ## 2 questions on area between curves! help please!

Ahh, yikes. I just typed out all of the images to prevent any further mishaps hah.

Find c > 0 such that the area of the region enclosed by the parabolas
y = x^2 - c^2 and y = c^2 - x^2 is 30.

Find the area of the region bounded by the parabola y=4x^2, the tangent line to this parabola at (3,36) and the x axis.

Ok, so my method of solving the first one was kind of strange, and I'm fairly certain that I used a completely incorrect method, but here it is...
I assumed that if I found the area in two quadrants then I could just multiply it by 2 and get the total area. So I did y = x^2 - c^2 = area of 15. Then I integrated, plugged in -c and c for x and solved but I came up with a wrong answer, which I think was something like 2.83.

For the second problem I found the began by finding the derivative which was 8x then found the equation for the line by plugging in 36, 3, 8 into y=mx+b equation, which ended up being y=8x+12. Then I integrated
S 8x+12 - 4x^2 and I'm not sure what to even use as the limits at this point, but I made multiple guesses and was wrong on all of them.

2. Originally Posted by sgares
Find such that the area of the region enclosed by the parabolas and is .

Find the area of the region bounded by the parabola , the tangent line to this parabola at and the axis.

Ok, so my method of solving the first one was kind of strange, and I'm fairly certain that I used a completely incorrect method, but here it is...
I assumed that if I found the area in two quadrants then I could just multiply it by 2 and get the total area. So I did = area of 15. Then I integrated, plugged in -c and c for x and solved but I came up with a wrong answer, which I think was something like 2.83.

For the second problem I found the began by finding the derivative which was 8x then found the equation for the line by plugging in 36, 3, 8 into y=mx+b equation, which ended up being y=8x+12. Then I integrated
S 8x+12 - 4x^2 and I'm not sure what to even use as the limits at this point, but I made multiple guesses and was wrong on all of them.

3. Ahh, yikes. I just typed out all of the images to prevent any further mishaps hah.

Find c > 0 such that the area of the region enclosed by the parabolas
y = x^2 - c^2 and y = c^2 - x^2 is 30.

Find the area of the region bounded by the parabola y=4x^2, the tangent line to this parabola at (3,36) and the x axis.

Ok, so my method of solving the first one was kind of strange, and I'm fairly certain that I used a completely incorrect method, but here it is...
I assumed that if I found the area in two quadrants then I could just multiply it by 2 and get the total area. So I did y = x^2 - c^2 = area of 15. Then I integrated, plugged in -c and c for x and solved but I came up with a wrong answer, which I think was something like 2.83.

For the second problem I found the began by finding the derivative which was 8x then found the equation for the line by plugging in 36, 3, 8 into y=mx+b equation, which ended up being y=8x+12. Then I integrated
S 8x+12 - 4x^2 and I'm not sure what to even use as the limits at this point, but I made multiple guesses and was wrong on all of them.

4. Originally Posted by sgares
Ahh, yikes. I just typed out all of the images to prevent any further mishaps hah.

Find c > 0 such that the area of the region enclosed by the parabolas
y = x^2 - c^2 and y = c^2 - x^2 is 30.

Find the area of the region bounded by the parabola y=4x^2, the tangent line to this parabola at (3,36) and the x axis.

Ok, so my method of solving the first one was kind of strange, and I'm fairly certain that I used a completely incorrect method, but here it is...
I assumed that if I found the area in two quadrants then I could just multiply it by 2 and get the total area. So I did y = x^2 - c^2 = area of 15. Then I integrated, plugged in -c and c for x and solved but I came up with a wrong answer, which I think was something like 2.83.

For the second problem I found the began by finding the derivative which was 8x then found the equation for the line by plugging in 36, 3, 8 into y=mx+b equation, which ended up being y=8x+12. Then I integrated
S 8x+12 - 4x^2 and I'm not sure what to even use as the limits at this point, but I made multiple guesses and was wrong on all of them.

The first one is formally unanswerable, but informally

Let $\displaystyle c^2-x^2<x^2-c^2\quad\forall{x}\in(a,b)$

Now then you must find c by solving the following(Easy) integral equation

$\displaystyle \int_a^{b}\bigg[(x^2-c^2)-(c^2-x^2)\bigg]dx=30$

The tangent line is given by

$\displaystyle y-36=24(x-3)$

Now let $\displaystyle g(x)=24(x-3)+36$

and $\displaystyle f(x)=4x^2$

Now find the two values of x such that

$\displaystyle f(x)=g(x)$

Then test one value in each to show that either

$\displaystyle \forall{x}\in(a,b)\quad{f(x)}<g(x)\text{ or }g(x)<f(x)$

Then the area would be

$\displaystyle A=\int_a^{b}g(x)-f(x)dx$

or

$\displaystyle A=\int_a^{b}f(x)-g(x)dx$

respectively

Good luck, and if you have any more questions, dont hesitate to ask.

Mathstud

5. Thank you for the help, I really do appreciate it! I'll get back to working on them now! =)