What is the arclength of #(t-3,t^2)# on #t in [1,2]#?
The arclength is
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 parametric equations (x(t) = t - 3) and (y(t) = t^2) on the interval ([1, 2]), 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 ]
Substituting the given parametric equations, we have:
[ L = \int_{1}^{2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt ]
[ L = \int_{1}^{2} \sqrt{(1)^2 + (2t)^2} dt ]
[ L = \int_{1}^{2} \sqrt{1 + 4t^2} dt ]
Now, we integrate:
[ L = \int_{1}^{2} \sqrt{1 + 4t^2} dt = \left[\frac{1}{4}t\sqrt{1 + 4t^2} + \frac{1}{8}\ln\left(2t + \sqrt{1 + 4t^2}\right)\right]_{1}^{2} ]
[ L = \left(\frac{1}{4}(2)\sqrt{1 + 4(2)^2} + \frac{1}{8}\ln\left(2(2) + \sqrt{1 + 4(2)^2}\right)\right) - \left(\frac{1}{4}(1)\sqrt{1 + 4(1)^2} + \frac{1}{8}\ln\left(2(1) + \sqrt{1 + 4(1)^2}\right)\right) ]
[ L = \left(\frac{1}{4}(2)\sqrt{17} + \frac{1}{8}\ln\left(4 + \sqrt{17}\right)\right) - \left(\frac{1}{4}\sqrt{5} + \frac{1}{8}\ln\left(2 + \sqrt{5}\right)\right) ]
[ L \approx 3.964 ]
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.
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 write the parametric equations represent the ellipse given by #x^2/9 + y^2/81=1#?
- What is the arc length of #f(t)=(t^2-4t,5-1/t) # over #t in [3,4] #?
- What is the arclength of #f(t) = (t-2,1/(t-2))# on #t in [0,1]#?
- How do you differentiate the following parametric equation: # (tsec(t/3-pi),t cos(pi-t))#?
- What is the derivative of #f(t) = (t/(t+2) , t/(t^2-1) ) #?
- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7