Find an arc length parametrization of r(t) = ⟨e^(t)sint, e^(t)cost, 8e^(t)⟩
To find the arc length parametrization of the curve defined by the vector function r(t) = ⟨e^(t)sint, e^(t)cost, 8e^(t)⟩, we first need to calculate the arc length integral.
The arc length of a curve parameterized by r(t) = ⟨f(t), g(t), h(t)⟩ with t between a and b, is given by the formula:
L = ∫[a,b] √[f'(t)^2 + g'(t)^2 + h'(t)^2] dt
Let's find the derivatives of each component of r(t):
r'(t) = ⟨e^(t)cost, -e^(t)sint, 8e^(t)⟩
Now, we'll calculate the magnitude of r'(t):
|r'(t)| = √[e^(2t)cos^2(t) + e^(2t)sin^2(t) + (8e^(t))^2]
= √[e^(2t)(cos^2(t) + sin^2(t)) + 64e^(2t)]
= √[e^(2t) + 64e^(2t)]
= √[65e^(2t)]
Since we want to find the arc length from t=a to t=b, we can write the integral as:
L = ∫[a,b] √[65e^(2t)] dt
To simplify the integration, we observe that 65 is a constant, so we can bring it outside the square root:
L = √65 ∫[a,b] e^(t) dt
Integrating e^t gives us:
L = √65 [e^t] |[a,b]
L = √65 (e^b - e^a)
Therefore, the arc length of the curve from r(a) to r(b) is √65 (e^b - e^a).
Now, if we want to obtain the arc length parametrization, we need to solve the equation L(t) = √65 (e^t - e^a) for t in terms of arc length L. This can be done by isolating t:
L/√65 = e^t - e^a
e^t = L/√65 + e^a
t = ln(L/√65 + e^a)
Hence, the arc length parametrization for the curve r(t) = ⟨e^(t)sint, e^(t)cost, 8e^(t)⟩ is given by:
r(L) = ⟨e^(ln(L/√65 + e^a))sin(ln(L/√65 + e^a)), e^(ln(L/√65 + e^a))cos(ln(L/√65 + e^a)), 8e^(ln(L/√65 + e^a))⟩
Simplifying further:
r(L) = ⟨(L/√65 + e^a) sin(ln(L/√65 + e^a)), (L/√65 + e^a) cos(ln(L/√65 + e^a)), 8(L/√65 + e^a)⟩
That is the arc length parametrization of the curve r(t).