Arc length of parametric curves practice problems
Arc length of a parametric curve: ∫ √((dx/dt)² + (dy/dt)²) dt. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Find the length of the curve \( \displaystyle x = 3 t - 3 \sin{\left(t \right)} \), \( \displaystyle y = 3 - 3 \cos{\left(t \right)} \), \( \displaystyle 0 \le t \le 2 \pi \).
Find the length of the curve \( \displaystyle x = e^{t} \cos{\left(t \right)} \), \( \displaystyle y = e^{t} \sin{\left(t \right)} \), \( \displaystyle 0 \le t \le 1 \).
Find the length of the curve \( \displaystyle x = 6 t^{2} \), \( \displaystyle y = 4 t^{3} \), \( \displaystyle 0 \le t \le 1 \).
Find the length of the curve \( \displaystyle x = e^{t} \cos{\left(t \right)} \), \( \displaystyle y = e^{t} \sin{\left(t \right)} \), \( \displaystyle 0 \le t \le 2 \).
Find the length of the curve \( \displaystyle x = t - \sin{\left(t \right)} \), \( \displaystyle y = 1 - \cos{\left(t \right)} \), \( \displaystyle 0 \le t \le 2 \pi \).
Find the length of the curve \( \displaystyle x = 2 \cos{\left(t \right)} \), \( \displaystyle y = 2 \sin{\left(t \right)} \), \( \displaystyle 0 \le t \le \pi \).
Find the length of the curve \( \displaystyle x = 4 \cos{\left(t \right)} \), \( \displaystyle y = 4 \sin{\left(t \right)} \), \( \displaystyle 0 \le t \le \pi \).
Find the length of the curve \( \displaystyle x = 2 t + 1 \), \( \displaystyle y = 2 t - 2 \), \( \displaystyle 0 \le t \le 4 \).
Find the length of the curve \( \displaystyle x = 4 \cos{\left(t \right)} \), \( \displaystyle y = 4 \sin{\left(t \right)} \), \( \displaystyle 0 \le t \le \frac{\pi}{2} \).
Find the length of the curve \( \displaystyle x = 1 - 3 t \), \( \displaystyle y = - 5 t - 2 \), \( \displaystyle 0 \le t \le 4 \).