# What is the arclength of #(e^(2t)-t,t-t/e^(t-1))# on #t in [-1,1]#?

The arc length formula for a parametric equation is:

Now plug into the formula and solve:

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

To find the arc length of the curve described by the parametric equations ( x(t) = e^{2t} - t ) and ( y(t) = t - \frac{t}{e^{t-1}} ) on the interval ( t ) in ([-1,1]), we use the formula:

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

Where:

- ( \frac{dx}{dt} ) represents the derivative of ( x(t) ) with respect to ( t )
- ( \frac{dy}{dt} ) represents the derivative of ( y(t) ) with respect to ( t )
- ( a ) and ( b ) are the lower and upper limits of the parameter ( t ), respectively.

By calculating the derivatives, we get:

[ \frac{dx}{dt} = 2e^{2t} - 1 ] [ \frac{dy}{dt} = 1 - \frac{1}{e^{t-1}} + \frac{t}{e^{t-1}} ]

Now, we substitute these derivatives into the arc length formula and integrate over the interval ([-1,1]):

[ L = \int_{-1}^{1} \sqrt{(2e^{2t} - 1)^2 + \left(1 - \frac{1}{e^{t-1}} + \frac{t}{e^{t-1}}\right)^2} dt ]

This integral needs to be solved numerically, as it does not have a simple closed form solution. You can use numerical integration techniques such as the trapezoidal rule or Simpson's rule to approximate the value of the integral.

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.

- How do you find the equation of a line tangent to the curve at point #t=-1# given the parametric equations #x=t^3+2t# and #y=t^2+t+1#?
- How do you differentiate the following parametric equation: # x(t)=lnt-2t, y(t)= cos^2t #?
- What is the derivative of #f(t) = (t^3-te^(1-t) , t^2-6t+e^t ) #?
- How do you differentiate the following parametric equation: # x(t)=te^t-lnt, y(t)= t^3e^t-t^2+t#?
- How do you find d/dt given #y=t^3 - 3t# and #x=t^2#?

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