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Worksheet: Arc Length by Integration

In this worksheet, we will practice setting up the integral that gives the arc length of the smooth curve defined as y=f(x) between two points.

Q1:

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

  • A 2 ο€½ π‘˜ 2  s i n
  • B a r c s i n ο€½ π‘˜ 2 
  • C s i n ο€½ π‘˜ 2 
  • D 2 ο€½ π‘˜ 2  a r c s i n
  • E 2 ( π‘˜ ) a r c s i n

Q2:

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.

Q3:

Let 𝐹 ( π‘₯ ) = ο€» π‘₯ , √ 9 βˆ’ π‘₯  2 on the interval [ 0 , 3 ] . By setting the arc length function 𝑠 ( 𝑑 ) to be the arc length between 𝐹 ( 0 ) and 𝐹 ( 𝑑 ) , find the coordinates of the point 𝑃 on this curve such that the arc length from 𝐹 ( 0 ) to 𝑃 is 1. Give your answer to 4 decimal places.

  • A ( 0 . 9 8 1 6 , 8 . 0 3 6 5 )
  • B ( 1 , 2 . 8 2 8 4 )
  • C ( 0 . 0 1 7 5 , 2 . 9 9 9 )
  • D ( 0 . 9 8 1 6 , 2 . 8 3 4 9 )
  • E ( 0 . 3 2 7 2 , 2 . 9 8 2 1 )

Q4:

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

Q5:

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?

Q6:

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 ) .

  • A 𝑓 ( 𝑠 ) = 2 ο€» 1 +  βˆ’ 3 3   οŠͺ  
  • B 𝑓 ( 𝑠 ) = 𝑠 βˆ’ 4 9  
  • C 𝑓 ( 𝑠 ) = 4 ο€» 1 +  βˆ’ 9 9   οŠͺ  
  • D 𝑓 ( 𝑠 ) = ( 2 7 𝑠 + 8 ) βˆ’ 4 9  
  • E 𝑓 ( 𝑠 ) = ( 1 8 𝑠 + 8 ) βˆ’ 8 2 7  

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

  • A π‘₯ = ( 2 7 𝑠 + 8 ) βˆ’ 4 9   , 𝑦 = ο„‘ ο„£ ο„£ ο„£ ο„   ( 2 7 𝑠 + 8 ) βˆ’ 4 9    
  • B 𝑓 ( 𝑠 ) = 4 ο€» 1 +  βˆ’ 9 9   οŠͺ   , 𝑦 = ο„‘ ο„£ ο„£ ο„£ ο„£ ο„£ ο„  βŽ› ⎜ ⎜ ⎝ 4 ο€» 1 +  βˆ’ 9 9 ⎞ ⎟ ⎟ ⎠   οŠͺ   
  • C 𝑓 ( 𝑠 ) = 2 ο€» 1 +  βˆ’ 3 3   οŠͺ   , 𝑦 = ο„‘ ο„£ ο„£ ο„£ ο„£ ο„£ ο„  βŽ› ⎜ ⎜ ⎝ 2 ο€» 1 +  βˆ’ 3 3 ⎞ ⎟ ⎟ ⎠   οŠͺ   
  • D π‘₯ = 𝑠 βˆ’ 4 9   , 𝑦 = ο„‘ ο„£ ο„£ ο„£ ο„   𝑠 βˆ’ 4 9    
  • E 𝑓 ( 𝑠 ) = ( 1 8 𝑠 + 8 ) βˆ’ 8 2 7   , 𝑦 = ο„‘ ο„£ ο„£ ο„£ ο„   ( 1 8 𝑠 + 8 ) βˆ’ 8 2 7    

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.

  • A ( 1 . 2 9 6 , 1 . 4 7 6 )
  • B ( 1 . 2 9 5 , 1 . 5 0 0 )
  • C ( 1 . 3 1 0 , 1 . 5 0 0 )
  • D ( 1 . 2 9 5 , 1 . 4 7 4 )

Q7:

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

Q8:

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

  • A πœ‹ 2
  • B πœ‹
  • C πœ‹ 4
  • D 2 πœ‹
  • E 4 πœ‹

Q9:

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.

  • A3.46410
  • B1.57080
  • C1.46410
  • D3.14159
  • E5.46410

Q10:

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.

  • A3.389
  • B 6 1 1 7
  • C 2 2 4 9
  • D 6 1 1 8
  • E 3 1 9

Q11:

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

  • A 𝑠 ( π‘₯ ) = 2 ο€» 1 +  βˆ’ 3 3 9 π‘₯ 4 3 2
  • B 𝑠 ( π‘₯ ) = ( 9 π‘₯ + 4 ) 2 7 3 2
  • C 𝑠 ( π‘₯ ) = 4 ο€» 1 +  βˆ’ 9 9 9 π‘₯ 4 3 2
  • D 𝑠 ( π‘₯ ) = ( 9 π‘₯ + 4 ) βˆ’ 8 2 7 3 2
  • E 𝑠 ( π‘₯ ) = ( 1 8 π‘₯ + 8 ) βˆ’ 8 2 7 3 2

Q12:

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.

Q13:

Perform the following.

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

  • A 2 𝑒 √ 𝑒 + 1 2 π‘₯ 2 π‘₯
  • B 1 √ 𝑒 + 1 2 π‘₯
  • C 2 √ 𝑒 + 1 2 π‘₯
  • D 𝑒 √ 𝑒 + 1 2 π‘₯ 2 π‘₯
  • E 𝑒 𝑒 + 1 2 π‘₯ 2 π‘₯

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

  • A βˆ’ 1 √ 𝑒 + 1 2 π‘₯
  • B βˆ’ 𝑒 √ 𝑒 + 1 2 π‘₯ 2 π‘₯
  • C 𝑒 √ 𝑒 + 1 2 π‘₯ 2 π‘₯
  • D 1 √ 𝑒 + 1 2 π‘₯
  • E βˆ’ 𝑒 √ 𝑒 + 1 2 π‘₯ 2 π‘₯

Hence, find ο„Έ √ 𝑒 + 1 2 π‘₯ .

  • A √ 𝑒 + 1 βˆ’ ο€» √ 𝑒 + 1  + 𝐢 2 π‘₯ βˆ’ 1 2 π‘₯ t a n h
  • B √ 𝑒 + 1 + ο€» √ 𝑒 + 1  + 𝐢 2 π‘₯ βˆ’ 1 2 π‘₯ t a n h
  • C 2 √ 𝑒 + 1 βˆ’ ο€» √ 𝑒 + 1  + 𝐢 2 π‘₯ βˆ’ 1 2 π‘₯ t a n h
  • D 2 √ 𝑒 + 1 + ο€» √ 𝑒 + 1  + 𝐢 2 π‘₯ βˆ’ 1 2 π‘₯ t a n h

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 .

Q14:

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

  • A ο„Έ √ 1 βˆ’ π‘₯ π‘₯ πœ‹ 0 2 c o s d
  • B ο„Έ  1 + π‘₯ π‘₯ πœ‹ 0 2 s i n d
  • C ο„Έ √ 1 βˆ’ π‘₯ π‘₯ πœ‹ 0 2 s i n d
  • D ο„Έ √ 1 + π‘₯ π‘₯ πœ‹ 0 2 c o s d
  • E ο„Έ √ 1 + π‘₯ π‘₯ πœ‹ 0 c o s d

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