Final answer:
To find the length of the curve, integrate the magnitude of the velocity function of the curve. The length of the curve is the integral of the magnitude of the velocity. Substitute the limits of integration to find the length.
Step-by-step explanation:
To find the length of the curve, we need to find the integral of the velocity function. The given curve is given by r(t) = 2t i + et j + e-t k. The velocity function can be found by differentiating the curve with respect to t.
v(t) = r'(t) = 2 i + e j - e-t k
The length of the curve is given by the integral of the magnitude of the velocity.
length = ∫|v(t)| dt = ∫√(4 + e^2 + e^(-2)) dt
By substituting the limits of integration (0 to 7) into the integral, we can find the length of the curve.