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Find the length of the curve.
r(t) = cos(3t)i+sin(3t)j+3 ln cos t k, 0 ≤ t ≤ π/4

User Tsionyx
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1 Answer

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Final answer:

To find the length of the curve, we can use the arc length formula and integrate the magnitude of the derivative of the position vector over the given range.

Step-by-step explanation:

To find the length of the curve, we can use the arc length formula:

L = ∫|r'(t)| dt

where r(t) is the position vector and r'(t) is the derivative of r(t) with respect to t. In this case, r(t) = cos(3t)i + sin(3t)j + 3 ln(cos(t))k. Taking the derivative of r(t), we get r'(t) = -3sin(3t)i + 3cos(3t)j - 3tan(t)ln(cos(t))i - 3ln(cos(t))sin(t)j. Now, we can find the magnitude of r'(t) which will give us the length of the curve:

|r'(t)| = sqrt((-3sin(3t))^2 + (3cos(3t))^2 + (-3tan(t)ln(cos(t)))^2 + (-3ln(cos(t))sin(t))^2). Integrate this expression over the given range 0 ≤ t ≤ π/4 to find the length of the curve.

User Kovica
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