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In this lesson, we will learn how to find the arc length of a parametrically defined curve.

Q1:

Calculate the arc length of over the given interval .

Q2:

Find the length of the curve with parametric equations 𝑥 = 3 𝑡 − 3 𝑡 c o s c o s and 𝑦 = 3 𝑡 − 3 𝑡 s i n s i n , where 0 ≤ 𝑡 ≤ 𝜋 .

Q3:

Express the length of the curve with parametric equations 𝑥 = 𝑡 − 2 𝑡 s i n and 𝑦 = 1 − 2 𝑡 c o s , where 0 ≤ 𝑡 ≤ 4 𝜋 , as an integral.

Q4:

Find the length of the curve with parametric equations 𝑥 = 2 𝑡 s i n − 1 and 𝑦 = 1 − 𝑡 l n 2 , where 0 ≤ 𝑡 ≤ 1 2 .

Q5:

Find the length of the curve with parametric equations 𝑥 = 𝑡 𝑡 s i n and 𝑦 = 𝑡 𝑡 c o s , where 0 ≤ 𝑡 ≤ 1 .

Q6:

Express the length of the curve with parametric equations 𝑥 = 𝑡 + 𝑒 − 𝑡 and 𝑦 = 𝑡 − 𝑒 − 𝑡 , where 0 ≤ 𝑡 ≤ 2 , as an integral.

Q7:

Find the length of the curve with parametric equations 𝑥 = 𝑒 − 𝑡 𝑡 and 𝑦 = 4 𝑒 𝑡 2 , where 0 ≤ 𝑡 ≤ 2 .

Q8:

Express the length of the curve with parametric equations 𝑥 = 𝑡 + √ 𝑡 and 𝑦 = 𝑡 − √ 𝑡 , where 0 ≤ 𝑡 ≤ 1 , as an integral.

Q9:

The position of a particle at time 𝑡 is 𝑡 , 𝑡 s i n c o s 2 2 . Find the distance the particle travels between 𝑡 = 0 and 𝑡 = 3 𝜋 .

Q10:

Find the length of the curve with parametric equations 𝑥 = 1 + 3 𝑡 2 and 𝑦 = 4 + 2 𝑡 3 , where 0 ≤ 𝑡 ≤ 1 .

Q11:

Find the length of the curve with parametric equations 𝑥 = 𝑡 2 and 𝑦 = 1 3 𝑡 3 , where 0 ≤ 𝑡 ≤ 1 .

Q12:

Find the length of the astroid with parametric equations 𝑥 = 𝑎 𝜃 c o s 3 and 𝑦 = 𝑎 𝜃 s i n 3 , where 𝑎 > 0 .

Q13:

Find the length of one arch of the cycloid with parametric equations 𝑥 = 𝑟 ( 𝑡 − 𝑡 ) s i n and 𝑦 = 𝑟 ( 1 − 𝑡 ) c o s .

Q14:

Consider the parametric equations 𝑥 = 𝑎 𝜃 c o s and 𝑦 = 𝑎 𝜃 s i n for 0 ≤ 𝜃 ≤ 2 𝜋 .

Express the arclength of this curve as an integral.

Evaluate the integral.

Q15:

Express the length of the curve with parametric equations 𝑥 = 𝑡 − 𝑡 2 and 𝑦 = 𝑡 4 , where 1 ≤ 𝑡 ≤ 4 , as an integral.

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