# area and length

• Jul 15th 2009, 10:07 AM
latavee
area and length
How are these two equations equal: (one interpreted as area and the other length)

2integral{-1 to 1} sqrt[1-x^2]dx and integral{-1 to 1} 1/sqrt[1-x^2]dx

First I understand that the second equation if a circle of radius 1 centered at the origin...and the first equation I'm considering the semicircle int he half plane...but why would they be equal? If I remove the 2 infront of the first integral, would it be the area of the other's region? confused!
• Jul 15th 2009, 10:23 AM
skeeter
Quote:

Originally Posted by latavee
How are these two equations equal: (one interpreted as area and the other length)

2integral{-1 to 1} sqrt[1-x^2]dx and integral{-1 to 1} 1/sqrt[1-x^2]dx

First I understand that the second equation if a circle of radius 1 centered at the origin...and the first equation I'm considering the semicircle int he half plane...but why would they be equal? If I remove the 2 infront of the first integral, would it be the area of the other's region? confused!

$2\int_{-1}^1 \sqrt{1-x^2} \, dx = \pi$ ... (unit circle area)

$\int_{-1}^1 \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin(1) - \arcsin(-1) = \frac{\pi}{2} - \left(-\frac{\pi}{2}\right) = \pi$

can't see how this is an arc length.

just looks like two definite integrals that yield the same numerical values, that's all.
• Jul 15th 2009, 10:29 AM
latavee
Ahh! I see, Thank you!