# What is the arclength of #f(t) = (-(t+3)^2,3t-4)# on #t in [0,1]#?

Arclength is given by:

This is a known integral. If you do not have it memorized apply integration by parts or look it up in a table of integrals:

Reverse the substitution:

Insert the limits of integration:

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

To find the arc length of the curve described by the function ( f(t) = (-(t+3)^2, 3t-4) ) on the interval ( t ) in ([0,1]), you can use the arc length formula:

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

Where ( a ) and ( b ) are the lower and upper limits of the parameter ( t ), and ( dx/dt ) and ( dy/dt ) are the derivatives of ( x ) and ( y ) with respect to ( t ), respectively.

First, find the derivatives:

[ dx/dt = -2(t+3) ] [ dy/dt = 3 ]

Next, plug these derivatives into the arc length formula and integrate over the interval ([0,1]):

[ L = \int_{0}^{1} \sqrt{(-2(t+3))^2 + (3)^2} , dt ]

[ = \int_{0}^{1} \sqrt{4(t+3)^2 + 9} , dt ]

[ = \int_{0}^{1} \sqrt{4t^2 + 24t + 25} , dt ]

This integral can be solved using standard calculus techniques, such as trigonometric substitution or completing the square.

Once you have found the antiderivative and evaluated it at the upper and lower limits, you will have the arc length of the curve on the interval ([0,1]).

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.

- What is the derivative of #f(t) = (tlnt, 3t^2-t ) #?
- What is the arc length of the curve given by #r(t)= (1/t,1,t)# on # t in [0, 1]#?
- What is the arc length of the curve given by #r(t)= (9sqrt(2),e^(9t),e^(-9t))# on # t in [3,4]#?
- How do you differentiate the following parametric equation: # (t-t^3,3/t^4-t^3)#?
- A soccer ball kicked at the goal travels in a path given by the parametric equations: x=50t; #y=-16t^2+32t#, At what two times will the ball be at a height of 6ft.?

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