How do you find the length of the curve defined by the polar equations r(t) = 7t and θ(t) = 3*cos(t) + 3*sin(t) + 2*t^2?

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The length of a curve defined in polar coordinates can be calculated using the formula:

L = ∫ab √( (dr/dt)^2 + (r^2) ) dt

Where:

  • r(t) is the radius as a function of t,
  • Theta (θ) is the angle related to the radius,
  • dr/dt is the derivative of r with respect to t,
  • L is the length of the curve from t=a to t=b.

To find the length of the specified curve:

  1. Differentiate r(t): r(t) = 7t
  2. Calculate dr/dt: dr/dt = 7
  3. Express r(t) in terms of t: r(t) = 7t
  4. Now, substitute these values into the length formula:
    5. L = ∫ab √( (7)^2 + (7t)^2 ) dt
= ∫ab √ (49 + 49t^2) dt 
= ∫ab 7√(1 + t^2) dt
  5. Calculate the definite integral from the limits of integration you choose for t.

Finally, evaluating the integral will provide the length of the curve for the specified range.

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