## Description

I would like you to show work and answer all parts of the question. Please write in a very neat and organize way to understand. Showing work for each step is important.

For the practice question, I would like you to not only find the arc length BUT you have to find it again using the Distance formula, and showing work is required as well.

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## Explanation & Answer

The solutions are ready, please ask if something is unclear.

1) The arc length formula is

3

2

∫ √1 + (𝑦 ′ (𝑥)) 𝑑𝑥,

−1

which gives

3

3

∫ √1 + (2)2 𝑑𝑥 = ∫ √5𝑑𝑥 = √5(3 − (−1)) = 𝟒√𝟓.

−1

−1

2) Use distance formula: the first point is

(−1, 𝑦(−1)) = (−1, −7),

the second point is

(3, 𝑦(3)) = (3,1),

the distance between them is

2

2

√(3 − (−1)) + (1 − (−7)) = √42 + 82 = 4√1 + 22 = 4√5,

which is of course the same.

𝐴

Consider ∫2 𝑒 −9𝑝 𝑑𝑝 from the definition of an improper integral. The antiderivative of 𝑒 −9𝑝 is

1

obviously − 9 𝑒 −9𝑝 (+𝐶), so the proper integral is equal...