I need to know how to integrate 2sqrt(100-x^2) from -10 to 10 by hand. It suggests a u-substitution but no u seems to exist. Please help.
Bret Norvilitis
Orchard Park HS
I was never really that sure about substitution but i think this is how it works...
When it says to do a u-sub it means set x to be equal to some function containing u. This also means that you should change your limits as well.
In this case, set $\displaystyle x = 10\sin(u)$ and hence $\displaystyle dx = 10\cos(u)du$ then $\displaystyle x^2 = 100\sin^2(u)$.
So put this into your equation and you get...
$\displaystyle \int 2 \sqrt(100-x^2)dx = \int 2 \sqrt(100-100\sin^2(u))du = \int 2 \sqrt(100(1-\sin^2(u))) 10\cos(u)du$.
= $\displaystyle \int 2 \sqrt(100cos^2(u)) 10\cos(u)$.
= $\displaystyle \int 200cos^2(u)$