Easy question but I forgot how to do it

s3a

Nov 2008
624
5
Question:
Find
such that the area of the region enclosed by the parabolas
and
is
.

My problem probably lies with the limits of integration. When I get x^2 = c^2 x = c, I assumed the limits of integration would be -c and c but I can't justify it not to mention the answer is wrong.

Can someone please help me?

Any help would be greatly appreciated!
Thanks in advance!
 

skeeter

MHF Helper
Jun 2008
16,216
6,764
North Texas
Question:
Find
such that the area of the region enclosed by the parabolas
and
is
.

My problem probably lies with the limits of integration. When I get x^2 = c^2 x = c, I assumed the limits of integration would be -c and c but I can't justify it not to mention the answer is wrong.

Can someone please help me?

Any help would be greatly appreciated!
Thanks in advance!
\(\displaystyle x^2-c^2 < c^2-x^2\) for \(\displaystyle -c < x < c\)

\(\displaystyle 150 = \int_{-c}^c (c^2-x^2) - (x^2-c^2) \, dx\)

\(\displaystyle 150 = 4\int_0^c c^2-x^2 \, dx\)

\(\displaystyle \frac{75}{2} = \left[c^2 x - \frac{x^3}{3}\right]_0^c\)

\(\displaystyle \frac{75}{2} = c^3 - \frac{c^3}{3}\)

\(\displaystyle \frac{75}{2} = \frac{2c^3}{3}\)

\(\displaystyle \frac{225}{4} = c^3\)

\(\displaystyle c = \sqrt[3]{\frac{225}{4}}\)
 
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Jul 2007
894
298
New Orleans
Question:
Find
such that the area of the region enclosed by the parabolas
and
is
.

My problem probably lies with the limits of integration. When I get x^2 = c^2 x = c, I assumed the limits of integration would be -c and c but I can't justify it not to mention the answer is wrong.

Can someone please help me?

Any help would be greatly appreciated!
Thanks in advance!
already was answer
 
Last edited:
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Jan 2010
354
173
Question:
Find
such that the area of the region enclosed by the parabolas
and
is
.

My problem probably lies with the limits of integration. When I get x^2 = c^2 x = c, I assumed the limits of integration would be -c and c but I can't justify it not to mention the answer is wrong.

Can someone please help me?

Any help would be greatly appreciated!
Thanks in advance!
If you are still wondering about why the limits of integration are \(\displaystyle -c<x<c\), you of course know we set the two expressions equal:

\(\displaystyle x^2-c^2=c^2-x^2\)

\(\displaystyle \implies 2x^2 = 2c^2\)

\(\displaystyle \implies x^2 = c^2\)

\(\displaystyle \implies \sqrt{x^2} = \sqrt{c^2}\)

Now, here is where most people make a mistake (or try to take shortcuts). What is the square root of \(\displaystyle x^2\) ? It's \(\displaystyle |x|\), NOT \(\displaystyle x\). So our next step will be:

\(\displaystyle \implies |x| = c\)

\(\displaystyle \implies x = \pm c\)

By the way, the reason we can say that \(\displaystyle \sqrt{c^2}=c\) is because the problem states that \(\displaystyle c>0\).
 
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