# Thread: two Q's: finding center of mass of a bounded region, and finding length of curve

1. ## two Q's: finding center of mass of a bounded region, and finding length of curve

I'm working on two problems. I've some rather rough solutions worked out, but am pretty sure they are wrong. Please show me step-wise the correct way to solve the following problems:

1.) find the center of mass of a region bounded by y=x and y=1-2x^2.

2.) sketch and calculate the length of the polar curve y=theta^2 from theta=0 to theta=4pi.

2. 2.) sketch and calculate the length of the polar curve y=theta^2 from theta=0 to theta=4pi.
Shouldn't that be $r={\theta}^{2}$?.

Anyway, arc length in polar is given by $L=\int_{\alpha}^{\beta}\sqrt{r^{2}+\left(\frac{dr} {d{\theta}}\right)^{2}}d{\theta}$

$\int_{0}^{4\pi}\sqrt{{\theta}^{4}+4{\theta}^{2}}d{ \theta}$

$\int_{0}^{4\pi}\sqrt{{\theta}^{2}({\theta}^{2}+4)} d{\theta}$

$=\int_{0}^{4\pi}{\theta}\sqrt{{\theta}^{2}+4}d{\th eta}$

You could use trig sub to solve this by using the sub:

${\theta}=2tan(x), \;\ d{\theta}=2sec^{2}(x)dx$