Question

How do you do this question?

Answer 1

Answer:

33/16

Step-by-step explanation:

x = y⁴/8 + 1/(4y²), 1 ≤ y ≤ 2

dx/dy = y³/2 − 1/(2y³)

Arc length is:

s = ∫ ds

s = ∫ √(1 + (dx/dy)²) dy

s = ∫₁² √(1 + (y³/2 − 1/(2y³))²) dy

s = ∫₁² √(1 + y⁶/4 − ½ + 1/(4y⁶)) dy

s = ∫₁² √(½ + y⁶/4 + 1/(4y⁶)) dy

s = ∫₁² ½ √(2 + y⁶ + 1/y⁶) dy

s = ∫₁² ½ √(y³ + 1/y³)² dy

s = ∫₁² ½ (y³ + 1/y³) dy

s = ½ (y⁴/4 − 1/(2y²)) |₁²

s = ½ (16/4 − 1/8) − ½ (1/4 − 1/2)

s = 33/16