# 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 Simpsonβs 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 Simpsonβs rule with 4 subintervals of width 0.5, what is the estimated arc length to 5 decimal places?

Using Simpsonβs 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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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
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**Q14: **

Find the arc length of the cardioid with polar equation .