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

planke

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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.
 
Re: Evaluate the Integral

planke said:
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.
 
Re: Evaluate the Integral

02x1(x1)2dx\displaystyle \int_{0}^{2}x\sqrt{1-(x-1)^{2}}dx

One thing you could do is let u=x1,   du=dx,   x=u+1\displaystyle u=x-1, \;\ du=dx, \;\ x=u+1

11(u+1)1u2du\displaystyle \int_{-1}^{1}(u+1)\sqrt{1-u^{2}}du

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

π2π2[1+sin(t)sin2(t)sin3(t)]dt\displaystyle \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\left[1+sin(t)-sin^{2}(t)-sin^{3}(t)\right]dt

Now, continue.
 
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