University:
Johns Hopkins University
Course:
AS.110.601 | Algebra I
Academic year:

2024
Views:

374

Pages:

6
Author:

Cristian Santos

Question 1 Consider an object moving along the parametrized curve with equations: x(t) = eᵗ + e⁻ᵗ y(t) = eᵗ - e⁻ᵗ where t is in the time interval [0,4] seconds. Fill in the blanks: The maximum speed of the object on the time interval is 5.458 at time t = 4. The minimum speed of the object on the time interval is 0.991 at time t = ¼ ln 2. Question 2 Find the exact length of the curve: y = √(x - x²) + sin⁻¹√x Answer To find the exact length: Parameterize x = sin²u Derive dy/du and dx/du Combine: ( 𝑑 𝑠 / 𝑑 𝑢 ) 2 = ( 𝑑 𝑥 / 𝑑 𝑢 ) 2 + ( 𝑑 𝑦 / 𝑑 𝑢 ) 2 = 4 𝑐 𝑜 𝑠 2 𝑢 (ds/du) 2 =(dx/du) 2 +(dy/du) 2 =4cos 2 u Exact length of the curve: 2 Question 3 Evaluate the iterated integral by converting to polar coordinates: ∫₀² ∫₀√(2x - x²) xy dy dx Answer Convert to polar coordinates: 𝐼 = ∫ 0 ( 𝜋 / 2 ) ∫ 0 2 𝑟 3 𝑐 𝑜 𝑠 2 ( 𝜃 ) 𝑠 𝑖 𝑛 ( 𝜃 ) 𝑑 𝑟 𝑑 𝜃 I=∫ 0 ( π/2)∫ 0 2 r 3 cos 2 (θ)sin(θ)drdθ Solving: Result: 2/3 Question 4 Find the volume of the given solid. The tetrahedron enclosed by the coordinate planes and the plane 2x + y + z = 4. Answer The region is bounded by: ∫ 0 2 ∫ 0 ( 4 − 2 𝑥 ) ∫ 0 ( 4 − 2 𝑥 − 𝑦 ) 𝑑 𝑧 𝑑 𝑦 𝑑 𝑥 ∫ 0 2 ∫ 0 ( 4−2x)∫ 0 ( 4−2x−y)dzdydx Integrating gives the volume. Question 5 Find the exact length of the curve: Y = y⁴/8 + 1/(4y²), 1 ≤ y ≤ 2 Answer Using: 𝐿 = ∫ 1 2 1 + ( 𝑑 𝑌 𝑑 𝑦 ) 2 𝑑 𝑦 L=∫ 1 2 1+( dy dY ) 2 dy Simplify derivatives and solve: Result expressed in terms of y for final length.

Integrals #6 - Questions and Answers

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