# 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 between and .

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Q2:

Write the integral required to calculate the length of the sine curve between and . Do not evaluate it.

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Q3:

Find the arc length of the curve defined by the parametric equations and .

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Q4:

Find the arc length of the cardioid with polar equation .

Q5:

Calculate the arc length of the curve between and , giving your answer to 5 decimal places.

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

Q6:

Find the function which gives the length of the arc from to .

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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.

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Q8:

The figure shows the curve with marked points and .

Use the secant between and 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 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 with two iterations to estimate the arc length from to 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 Simpsonβs rule with 4 subintervals, the first summand is . What is the second summand in estimating the arc length? Give your answer to 4 decimal places.

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

Using Simpsonβ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 , with marked points , , , and .

Given that the arc length from to is given by , the first thing we need for an arc length parameterization is the inverse . Determine so that .

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Hence, give the arc length parameterization , of the curve.

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• B ,
• C ,
• D ,
• E ,

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

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Q11:

The exact length of the curve between and is 4.785154 to 6 decimal places. Estimate this using the trapezoidal rule with for your integral. Give your answer to 6 decimal places.

Q12:

Perform the following.

Find the derivative of .

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Find the derivative of .

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Hence, find .

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Use your results above to calculate, to 5 decimal places, the arc length of the curve between and .

Q13:

Work out the length of the arc between and . Give your answer as a fraction.

• A3.389
• B
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• D
• E

Q14:

Find the arc length of the cardioid with polar equation .