How do you determine the length of #x=3t^2#, #y=t^3+4t# for t is between [0,2]?

Answer 1

Hi, are you sure about the two parametric equations?

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To determine the length of the curve defined by (x = 3t^2) and (y = t^3 + 4t) for (t) between ([0, 2]), we use the formula for arc length of a parametric curve:

[L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt]

Where:

  • (a) and (b) are the lower and upper bounds of (t), respectively (in this case, (a = 0) and (b = 2)).
  • (\frac{dx}{dt}) and (\frac{dy}{dt}) are the derivatives of (x) and (y) with respect to (t), respectively.

First, find the derivatives of (x) and (y) with respect to (t):

[\frac{dx}{dt} = 6t] [\frac{dy}{dt} = 3t^2 + 4]

Now, plug these derivatives into the formula and integrate over the given range:

[L = \int_{0}^{2} \sqrt{(6t)^2 + (3t^2 + 4)^2} dt]

[L = \int_{0}^{2} \sqrt{36t^2 + 9t^4 + 24t^2 + 16} dt]

[L = \int_{0}^{2} \sqrt{9t^4 + 60t^2 + 16} dt]

This integral may not have a simple closed-form solution. It can be evaluated using numerical methods or specialized software.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7