Question
Find an arc length parameterization of the curve that has the same orientation as the given curve and for which the reference point corresponds to t=0. Use an arc length s as a parameter. r(t) = 3(e^t) cos (t)i + 3(e^t)sin(t)j; 0<=t<=(3.14/2)
135
likes676 views
Answer to a math question Find an arc length parameterization of the curve that has the same orientation as the given curve and for which the reference point corresponds to t=0. Use an arc length s as a parameter. r(t) = 3(e^t) cos (t)i + 3(e^t)sin(t)j; 0<=t<=(3.14/2)
Cristian
4.7119 Answers
The derivatives are x'(t) = \frac{d}{dt}(3e^tcost) = 3e^tcost - 3e^tsint = 3\sqrt{2}cos(t+\frac{\pi}{4})
y'(t) = \frac{d}{dt}(3e^tsint) = 3e^tsint + 3e^tcost = 3\sqrt{2}sin(t+\frac{\pi}{4})
So norm of the derivative is ||r'(t)|| = \sqrt{x'(t)^2 + y'(t)^2} = 3\sqrt{2}e^t
The arc length parametrization is s(t) = \int_0^t ||r'(\tau)|| d\tau x(t) = \int_0^t 3\sqrt{2}e^{\tau}d\tau = (3\sqrt{2}e^{\tau}]_0^t = 3\sqrt{2}(e^t-1)
So, the arc length parametrization is s(t) = 3\sqrt{2}(e^t-1)
Frequently asked questions (FAQs)
What is the radius of a sphere given its volume is 1256 cubic units?
+
What is the result of subtracting vector A (magnitude 10) from vector B (magnitude 15) and then adding vector C (magnitude 5)?
+
Question: What is the square root of 64?
+
New questions in Mathematics