Lesson: Arc Length by Integration

In this lesson, we will learn how to set up the integral that gives the arc length of the smooth curve defined as y=f(x) between two points.

Worksheet: Arc Length by Integration • 14 Questions

Q1:

Using a trigonometric substitution, determine the arc length of the curve 𝑦 = √ 4 βˆ’ π‘₯  between π‘₯ = 0 and π‘₯ = π‘˜ .

Q2:

Write the integral required to calculate the length of the sine curve between π‘₯ = 0 and π‘₯ = πœ‹ . Do not evaluate it.

Q3:

Find the arc length of the curve defined by the parametric equations π‘₯ = 𝑑 c o s and 𝑦 = 𝑑 s i n .

Q4:

Find the arc length of the cardioid with polar equation π‘Ÿ = 2 + 2 πœƒ c o s .

Q5:

Calculate the arc length of the curve 𝑦 = √ 4 βˆ’ π‘₯ 2 between π‘₯ = 0 and π‘₯ = 2 , giving your answer to 5 decimal places.

Q6:

Find the function 𝑠 ( π‘₯ ) which gives the length of the arc 𝑦 = √ π‘₯ 3 from ( 0 , 0 ) to ο€» π‘₯ , √ π‘₯  3 .

Q7:

Let on the interval . By setting the arc length function to be the arc length between and , find the coordinates of the point on this curve such that the arc length from to is 1. Give your answer to 4 decimal places.

Q8:

The figure shows the curve 𝑦 = 𝑒 + 𝑒 2 π‘₯ βˆ’ π‘₯ with marked points 𝐴 ( 1 , 1 . 5 4 3 ) and 𝐡 ( 2 , 3 . 7 6 2 ) .

Use the secant between 𝐴 and 𝐡 to get a lower bound on the length of the curve between those points. Give your answer to 3 decimal places.

Use the three additional points at π‘₯ = 1 . 2 5 , 1 . 5 , 1 . 7 5 to get a better approximation of this length to 3 decimal places.

Calculate the length of the curve exactly, giving your answer to 4 decimal places.

Q9:

The figure shows the graph of 𝑦 = π‘₯ 3  with two iterations to estimate the arc length from π‘₯ = 1 to π‘₯ = 3 using 2 and then 4 subintervals and adding lengths of the corresponding line segments.

A table of integrals gives the formula

Use this to compute the exact arc length to 5 decimal places.

Estimate the arc length using the 2 line segments in the first figure, giving your answer to 5 decimal places.

Estimate the arc length using the 4 line segments in the second figure, giving your answer to 5 decimal places.

Using Simpson’s rule with 4 subintervals, the first summand is 1 6 𝑓 ( 1 ) = √ 1 3 1 8 . What is the second summand in estimating the arc length? Give your answer to 4 decimal places.

Using Simpson’s rule with 4 subintervals of width 0.5, what is the estimated arc length to 5 decimal places?

Using Simpson’s rule with 8 subintervals of width 0.25, what is the estimated arc length to 5 decimal places?

Q10:

The figure shows a part of the curve 𝐹 ( π‘₯ ) = √ π‘₯  , with marked points 𝐴 ( 0 , 0 ) , 𝐡 , 𝐢 , and 𝐷 .

Given that the arc length from ( 0 , 0 ) to ( π‘₯ , 𝐹 ( π‘₯ ) ) is given by 𝑠 ( π‘₯ ) = ( 9 π‘₯ + 4 ) βˆ’ 8 2 7   , the first thing we need for an arc length parameterization is the inverse 𝑠 ( π‘₯ )   . Determine 𝑓 so that π‘₯ = 𝑓 ( 𝑠 ) .

Hence, give the arc length parameterization π‘₯ = 𝑓 ( 𝑠 ) , 𝑦 = 𝑔 ( 𝑠 ) of the curve.

The arc length between each of the points 𝐴 , 𝐡 , 𝐢 , and 𝐷 is one unit. Give the coordinates of 𝐢 to 3 decimal places.

Q11:

The exact length of the curve 𝑦 = 𝑒  between π‘₯ = 1 and π‘₯ = 2 is 4.785154 to 6 decimal places. Estimate this using the trapezoidal rule with 𝑛 = 1 0 for your integral. Give your answer to 6 decimal places.

Q12:

Perform the following.

Find the derivative of √ 𝑒 + 1 2 π‘₯ .

Find the derivative of t a n h βˆ’ 1 2 π‘₯ ο€» √ 𝑒 + 1  .

Hence, find ο„Έ √ 𝑒 + 1 2 π‘₯ .

Use your results above to calculate, to 5 decimal places, the arc length of the curve 𝑦 = 𝑒 π‘₯ between π‘₯ = 1 and π‘₯ = 3 .

Q13:

Work out the length of the arc 𝑦 = π‘₯ + 3 2 3 2 π‘₯   between π‘₯ = 1 and π‘₯ = 3 . Give your answer as a fraction.

Q14:

Find the arc length of the cardioid with polar equation π‘Ÿ = 1 + πœƒ s i n .

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