Find the length of the curve r(t) = 5t, 3 cos(t), 3 sin(t) , -3 ≤ t ≤ 3?

Solution:

r(t) = 5t, 3 cos(t), 3 sin(t) [Given]

We have to compare it with

r (t) = (x, y, z)

So x = 5t, y = 3 cos (t), z = 3 sin (t)

By using the arc length formula

\(L=\int_{a}^{b}\sqrt{(x')^{2}+(y')^{2}+(z')^{2}}dt\)

We know that the derivatives of x, y and z are:

x’ = 5

y’ = -3 sin t

z’ = 3 cos t

Now substitute these values

\(\\L=\int_{-3}^{3}\sqrt{(5)^{2}+(-3 sint)^{2}+(3cost)^{2}}dt \\ \\L=\int_{-3}^{3}\sqrt{(5)^{2}+9(1)}dt \\ \\L=\sqrt{34}\int_{-3}^{3}dt \\\\L=\sqrt{34}t |_{-3}^{3} \\L=\sqrt{34}[3-(-3)] \\L=6\sqrt{34}\)

Therefore, the length of the curve is 6√34

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Find the length of the curve r(t) = 5t, 3 cos(t), 3 sin(t) , -3 ≤ t ≤ 3?

Summary:

The length of the curve r(t) = 5t, 3 cos(t), 3 sin(t) , -3 ≤ t ≤ 3 is 6√34.