Evaluate int [0, 2] [x sqrt(1 - (x - 1)^2)] dx

[planke]

S (from 0 to 2) xsqrt(1-(x-1)^2)dx is the integral I am trying to evaluate. If anybody can show me how to get to the answer (pi/2) that would be really helpful. I keep getting zero for some reason.

 

[Deleted member 4993]

Re: Evaluate the Integral

S (from 0 to 2) xsqrt(1-(x-1)^2)dx is the integral I am trying to evaluate. If anybody can show me how to get to the answer (pi/2) that would be really helpful. I keep getting zero for some reason.

Please share with us your work - where you are getting zero - so that we know where to begin to help you.

 

[galactus]

Re: Evaluate the Integral

\(\displaystyle \int_{0}^{2}x\sqrt{1-(x-1)^{2}}dx\)

One thing you could do is let \(\displaystyle u=x-1, \;\ du=dx, \;\ x=u+1\)

\(\displaystyle \int_{-1}^{1}(u+1)\sqrt{1-u^{2}}du\)

Now, another sub may be in order. We can let \(\displaystyle u=sin(t), \;\ du=cos(t)dt\)

\(\displaystyle \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\left[1+sin(t)-sin^{2}(t)-sin^{3}(t)\right]dt\)

Now, continue.