How to Find the Length of the Curve given by r(t) = 5t, 3 cos(t), 3 sin(t), 5t^5?

To find the length of the curve defined by the vector function r(t) = (5t, 3 cos(t), 3 sin(t), 5t^5), we use the formula for the arc length of a space curve:

L = ∫ab ||r'(t)|| dt

Here, ||r'(t)|| represents the magnitude of the derivative of the vector function. To proceed, we’ll first differentiate r(t):

r'(t) = (d/dt[5t], d/dt[3 cos(t)], d/dt[3 sin(t)], d/dt[5t^5])
      = (5, -3 sin(t), 3 cos(t), 25t^4)

Next, we find the magnitude:

||r'(t)|| = sqrt[(5)^2 + (-3 sin(t))^2 + (3 cos(t))^2 + (25t^4)^2]
            = sqrt[25 + 9 sin^2(t) + 9 cos^2(t) + 625t^8]

Using the identity sin^2(t) + cos^2(t) = 1, we can simplify further:

||r'(t)|| = sqrt[25 + 9 + 625t^8]
            = sqrt[34 + 625t^8]

Now, we can integrate this with respect to t from a to b to determine the length of the curve:

L = ∫ab sqrt[34 + 625t^8] dt

Evaluating this integral will give you the length of the curve between the specified limits a and b. You might require numerical methods or advanced calculus techniques depending on the values you choose for a and b.