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Evaluate cfds\int_{\mathbf{c}} f\, ds, where f(x,y,z)=zf (x, y, z) = z and c(t)=(tcost,tsint,t)\mathbf{c}(t) = (t \cos t, t \sin t, t) for 0tt00 \leq t \leq t_0.


Step 1
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Curve c\mathbf{c} is parametrized by

{x(t)=tcost,y(t)=tsint,z(t)=t,0tt0.\begin{equation*} \left\{\begin{aligned} x(t)&=t\cos t,\\ y(t)&=t\sin t,\\ z(t)&=t, \end{aligned}\right.\quad 0\leq t\leq t_0. \end{equation*}

First, we compute c(t)||\mathbf{c}'(t)||:

c(t)=[dx(t)dt]2+[dy(t)dt]2+[dz(t)dt]2=[d(tcost)dt]2+[d(tsint)dt]2+[dtdt]2=(costtsint)2+(sint+tcost)2+12=cos2t+sin2t+t2(cos2t+sin2t)+1=1+t2+1=t2+2.\begin{align*} ||\mathbf{c}'(t)||&= \sqrt{\left[\dfrac{dx(t)}{dt}\right]^2+\left[\dfrac{dy(t)}{dt}\right]^2+\left[\dfrac{dz(t)}{dt}\right]^2}\\&=\sqrt{\left [\dfrac{d(t\cos t)}{dt}\right ]^2+\left [\dfrac{d(t\sin t)}{dt}\right ]^2+\left [\dfrac{dt}{dt}\right ]^2}\\ &=\sqrt{(\cos t-t\sin t)^2+(\sin t+t\cos t)^2+1^2}\\ &=\sqrt{\cos^2 t+\sin^2 t+t^2(\cos^2 t+\sin^2t)+1}\\ &=\sqrt{1+t^2+1}\\ &=\sqrt{t^2+2}. \end{align*}

Next, we substitute for xx, yy, zz in terms of tt to obtain

f(x,y,z)=z=t,\begin{equation*} f(x,y,z)=z=t, \end{equation*}

along c\mathbf{c}.

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