Worksheet: Arc Length by Integration

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In this worksheet, we will practice using integration to find the length of a curve.

Q1:

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

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

Q2:

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‘₯ οŽ„   s i n d
  • B ο„Έ √ 1 + π‘₯ π‘₯ οŽ„  c o s d
  • C ο„Έ √ 1 βˆ’ π‘₯ π‘₯ οŽ„   c o s d
  • D ο„Έ  1 + π‘₯ π‘₯ οŽ„   s i n d
  • E ο„Έ √ 1 + π‘₯ π‘₯ οŽ„   c o s d

Q3:

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

  • A3.14159
  • B1.57080
  • C5.46410
  • D3.46410
  • E1.46410

Q4:

Find the function 𝑠(π‘₯) which gives the length of the arc 𝑦=√π‘₯ from (0,0) to ο€»π‘₯,√π‘₯ο‡οŠ©.

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

Q5:

Let 𝐹(π‘₯)=ο€»π‘₯,√9βˆ’π‘₯ο‡οŠ¨ 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 ( 0 . 3 2 7 2 , 2 . 9 8 2 1 )
  • C ( 1 , 2 . 8 2 8 4 )
  • D ( 0 . 0 1 7 5 , 2 . 9 9 9 )
  • E ( 0 . 9 8 1 6 , 2 . 8 3 4 9 )

Q6:

The figure shows the curve 𝑦=𝑒+𝑒2ο—οŠ±ο— with marked points 𝐴(1,1.543) and 𝐡(2,3.762).

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‘₯=1.25,1.5,1.75 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.

Q7:

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 ο„Έβˆšπ‘Ž+𝑒𝑒=𝑒2βˆšπ‘Ž+𝑒+π‘Ž2𝑒+βˆšπ‘Ž+𝑒+.dlnC

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 16𝑓(1)=√1318. 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?

Q8:

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‘₯)=(9π‘₯+4)βˆ’827, 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 ) = ( 1 8 𝑠 + 8 ) βˆ’ 8 2 7  
  • B 𝑓 ( 𝑠 ) = 4 ο€» 1 +  βˆ’ 9 9   οŠͺ  
  • C 𝑓 ( 𝑠 ) = 2 ο€» 1 +  βˆ’ 3 3   οŠͺ  
  • D 𝑓 ( 𝑠 ) = 𝑠 βˆ’ 4 9  
  • E 𝑓 ( 𝑠 ) = ( 2 7 𝑠 + 8 ) βˆ’ 4 9  

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

  • A 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   
  • 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 π‘₯ = 𝑠 βˆ’ 4 9   , 𝑦 = ο„‘ ο„£ ο„£ ο„£ ο„   𝑠 βˆ’ 4 9    
  • D 𝑓 ( 𝑠 ) = ( 1 8 𝑠 + 8 ) βˆ’ 8 2 7   , 𝑦 = ο„‘ ο„£ ο„£ ο„£ ο„   ( 1 8 𝑠 + 8 ) βˆ’ 8 2 7    
  • E π‘₯ = ( 2 7 𝑠 + 8 ) βˆ’ 4 9   , 𝑦 = ο„‘ ο„£ ο„£ ο„£ ο„   ( 2 7 𝑠 + 8 ) βˆ’ 4 9    

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

Q9:

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=10 for your integral. Give your answer to 6 decimal places.

Q10:

Perform the following.

Find the derivative of βˆšπ‘’+1οŠ¨ο—.

  • A 2 𝑒 √ 𝑒 + 1    
  • B 𝑒 √ 𝑒 + 1    
  • C 𝑒 𝑒 + 1    
  • D 2 √ 𝑒 + 1  
  • E 1 √ 𝑒 + 1  

Find the derivative of tanhοŠ±οŠ§οŠ¨ο—ο€»βˆšπ‘’+1.

  • A 1 √ 𝑒 + 1  
  • B βˆ’ 1 √ 𝑒 + 1  
  • C βˆ’ 𝑒 √ 𝑒 + 1    
  • D 𝑒 √ 𝑒 + 1    
  • E βˆ’ 𝑒 √ 𝑒 + 1    

Hence, find ο„Έβˆšπ‘’+1οŠ¨ο—.

  • A 2 √ 𝑒 + 1 + ο€» √ 𝑒 + 1  + 𝐢       t a n h
  • B √ 𝑒 + 1 + ο€» √ 𝑒 + 1  + 𝐢       t a n h
  • C √ 𝑒 + 1 βˆ’ ο€» √ 𝑒 + 1  + 𝐢       t a n h
  • D 2 √ 𝑒 + 1 βˆ’ ο€» √ 𝑒 + 1  + 𝐢       t a n h

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

Q11:

Work out the length of the arc 𝑦=π‘₯+3232π‘₯ between π‘₯=1 and π‘₯=3. Give your answer as a fraction.

  • A 3 1 9
  • B 2 2 4 9
  • C3.389
  • D 6 1 1 7
  • E 6 1 1 8

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