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Calculus II · Parametric & Polar Curves

Parametric Arc Length — Practice Problems and Worked Solutions

Integrate speed along the parameter interval.

The formulas you need here

Parametric arc length

L = \int_{a}^{b} (\frac{d x}{d t})^{2} + (\frac{d y}{d t})^{2} d t

What it says: Pythagoras on an infinitesimal step, added along the curve.
When to reach for it: The length of a parametric path, or of y = f(x) with x as its own parameter.
Watch out: The square root covers the sum of both squares, and the limits are values of t, not of x.

L = \int_{0}^{2 π} sin^{2} t + cos^{2} t d t = 2 π

The unit circle — the identity collapses the root to 1.

16 worked problems

Every step is generated and checked by a computer algebra system.

1.
Find the arc length of the parametric curve.
$x = t, y = 2 t, 0 \leq t \leq 1$
Answer: $5$
Step-by-step solution
1. $L = \int_{a}^{b} (x^{'})^{2} + (y^{'})^{2} d t$
2. $L = \int_{0}^{1} 5 d t = 5$
2.
Find the arc length of the parametric curve.
$x = 3 t, y = 4 t, 0 \leq t \leq 2$
Answer: $10$
Step-by-step solution
1. $L = \int_{a}^{b} (x^{'})^{2} + (y^{'})^{2} d t$
2. $L = \int_{0}^{2} 5 d t = 10$
3.
Find the arc length of the parametric curve.
$x = t, y = t^{2}, 0 \leq t \leq 1$
Answer: $\frac{asinh ( 2 )}{4} + \frac{5}{2}$
Step-by-step solution
1. $L = \int_{a}^{b} (x^{'})^{2} + (y^{'})^{2} d t$
2. $L = \int_{0}^{1} 4 t^{2} + 1 d t = \frac{asinh ( 2 )}{4} + \frac{5}{2}$
4.
Find the arc length of the parametric curve.
$x = cos (t), y = sin (t), 0 \leq t \leq \frac{π}{2}$
Answer: $\frac{π}{2}$
Step-by-step solution
1. $L = \int_{a}^{b} (x^{'})^{2} + (y^{'})^{2} d t$
2. $L = \int_{0}^{\frac{π}{2}} 1 d t = \frac{π}{2}$
5.
Find the arc length of the parametric curve.
$x = 2 cos (t), y = 2 sin (t), 0 \leq t \leq π$
Answer: $2 π$
Step-by-step solution
1. $L = \int_{a}^{b} (x^{'})^{2} + (y^{'})^{2} d t$
2. $L = \int_{0}^{π} 2 d t = 2 π$
6.
Find the arc length of the parametric curve.
$x = t^{2}, y = \frac{2 t ^{3}}{3}, 0 \leq t \leq 1$
Answer: $- \frac{2}{3} + \frac{4 2}{3}$
Step-by-step solution
1. $L = \int_{a}^{b} (x^{'})^{2} + (y^{'})^{2} d t$
2. $L = \int_{0}^{1} 2 t^{4} + t^{2} d t = - \frac{2}{3} + \frac{4 2}{3}$
7.
Find the arc length of the parametric curve.
$x = t, y = 3 t, 0 \leq t \leq 2$
Answer: $210$
Step-by-step solution
1. $L = \int_{a}^{b} (x^{'})^{2} + (y^{'})^{2} d t$
2. $L = \int_{0}^{2} 10 d t = 210$
8.
Find the arc length of the parametric curve.
$x = 5 t, y = 12 t, 0 \leq t \leq 1$
Answer: $13$
Step-by-step solution
1. $L = \int_{a}^{b} (x^{'})^{2} + (y^{'})^{2} d t$
2. $L = \int_{0}^{1} 13 d t = 13$
9.
Find the arc length of the parametric curve.
$x = 8 t, y = 15 t, 0 \leq t \leq 2$
Answer: $34$
Step-by-step solution
1. $L = \int_{a}^{b} (x^{'})^{2} + (y^{'})^{2} d t$
2. $L = \int_{0}^{2} 17 d t = 34$
10.
Find the arc length of the parametric curve.
$x = 3 cos (t), y = 3 sin (t), 0 \leq t \leq \frac{π}{2}$
Answer: $\frac{3 π}{2}$
Step-by-step solution
1. $L = \int_{a}^{b} (x^{'})^{2} + (y^{'})^{2} d t$
2. $L = \int_{0}^{\frac{π}{2}} 3 d t = \frac{3 π}{2}$
11.
Find the arc length of the parametric curve.
$x = t, y = 4 t, 1 \leq t \leq 3$
Answer: $217$
Step-by-step solution
1. $L = \int_{a}^{b} (x^{'})^{2} + (y^{'})^{2} d t$
2. $L = \int_{1}^{3} 17 d t = 217$
12.
Find the arc length of the parametric curve.
$x = cos (t), y = sin (t), 0 \leq t \leq π$
Answer: $π$
Step-by-step solution
1. $L = \int_{a}^{b} (x^{'})^{2} + (y^{'})^{2} d t$
2. $L = \int_{0}^{π} 1 d t = π$
13.
Find the arc length of the parametric curve.
$x = 4 cos (t), y = 4 sin (t), 0 \leq t \leq 2 π$
Answer: $8 π$
Step-by-step solution
1. $L = \int_{a}^{b} (x^{'})^{2} + (y^{'})^{2} d t$
2. $L = \int_{0}^{2 π} 4 d t = 8 π$
14.
Find the arc length of the parametric curve.
$x = 6 t, y = 8 t, 0 \leq t \leq 3$
Answer: $30$
Step-by-step solution
1. $L = \int_{a}^{b} (x^{'})^{2} + (y^{'})^{2} d t$
2. $L = \int_{0}^{3} 10 d t = 30$
15.
Find the arc length of the parametric curve.
$x = 2 t, y = 5 t, 0 \leq t \leq 4$
Answer: $429$
Step-by-step solution
1. $L = \int_{a}^{b} (x^{'})^{2} + (y^{'})^{2} d t$
2. $L = \int_{0}^{4} 29 d t = 429$
16.
Find the arc length of the parametric curve.
$x = 5 cos (t), y = 5 sin (t), 0 \leq t \leq \frac{π}{3}$
Answer: $\frac{5 π}{3}$
Step-by-step solution
1. $L = \int_{a}^{b} (x^{'})^{2} + (y^{'})^{2} d t$
2. $L = \int_{0}^{\frac{π}{3}} 5 d t = \frac{5 π}{3}$

The formulas you need here

Parametric arc length

16 worked problems

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