# What is the arclength of #f(t) = (t^3-1,t^2-1)# on #t in [2,3]#?

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

To find the arc length of the curve defined by ( f(t) = (t^3 - 1, t^2 - 1) ) on the interval ( t ) in ([2,3]), we use the formula for arc length:

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

For the given function ( f(t) ), its derivatives are:

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

Now, substitute these derivatives into the arc length formula and integrate over the interval ([2,3]):

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

[ L = \int_{2}^{3} \sqrt{9t^4 + 4t^2} , dt ]

This integral is usually computed numerically since it doesn't have a simple closed-form solution.

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

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.

- For #f(t)= (te^(t-1),t^2-t+1)# what is the distance between #f(0)# and #f(2)#?
- How do you differentiate the following parametric equation: # x(t)=tlnt, y(t)= cost-tsin^2t #?
- How do you differentiate the following parametric equation: # x(t)=(t+3t^2)e^t , y(t)=t/e^(3t) #?
- What is the arclength of #f(t) = (t^2-5,e^t)# on #t in [1,3]#?
- Find the exact length of the curve (parametrics)?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7