Question

(a) Find the arc length of the parametrized curve
x= et cos t y= et z= et sin t for 0≤ t ≤2π

Answer 1

The parameterized curve is given by the formulas:

$x(t)=e^t cos(t); y(t)=e^t; z(t)=e^t sin(t)$

The formula for calculating the length of the curve given parametrically is as follows:

$L=\int{\sqrt{(x^{'})^2+(y^{'})^2+(z^{'})^2} dt}$

Let's calculate and simplify the integrand function: $x^{'}=e^t (cos(t) - sin(t)); y^{'}=e^t; z^{'}=e^t (sin(t)+cos(t))$

$\sqrt{(x^{'})^2+(y^{'})^2+(z^{'})^2}=e^t \sqrt{(cost-sint)^2+1+(sint+cost)^2}=e^t \sqrt{3}$

After that, the integration is carried out easily

$L=\int^{2\pi}_0 \sqrt{3} e^t dt=\sqrt{3} (e^{2\pi}-1)$

Answer: The arc length is $\sqrt{3} (e^{2\pi}-1)$