How do you do this question?

π/6 [37^(³/₂) − 1] ≈ 117.3187

Step-by-step explanation:

The surface area is:

S = ∫ 2π (x − 0) √(1 + (dx/dy)²) dy

0 ≤ x ≤ 3, so -4 ≤ y ≤ 5.

Find dx/dy.

y = 5 − x²

x² = 5 − y

x = √(5 − y)

dx/dy = ½ (5 − y)^(-½) (-1)

dx/dy = -½ (5 − y)^(-½)

(dx/dy)² = ¼ (5 − y)^(-1)

(dx/dy)² = 1 / (4 (5 − y))

Plug in:

S = ∫₋₄⁵ 2π x √(1 + 1 / (20 − 4y)) dy

S = ∫₋₄⁵ 2π √(5 − y) √(1 + 1 / (4 (5 − y))) dy

S = ∫₋₄⁵ 2π √((5 − y) + 1/4)) dy

S = ∫₋₄⁵ 2π √(5.25 − y) dy

If u = 5.25 − y, then du = -dy.

S = ∫ 2π √u (-du)

S = -2π ∫ √u du

S = -2π (⅔ u^(³/₂))

S = -4π/3 u^(³/₂)

Substitute back:

S = -4π/3 (5.25 − y)^(³/₂)

Evaluate between y=-4 and y=5.

S = [-4π/3 (5.25 − 5)^(³/₂)] − [-4π/3 (5.25 − -4)^(³/₂)]

S = -4π/3 (0.25)^(³/₂) + 4π/3 (9.25)^(³/₂)

S = π/6 [37^(³/₂) − 1]

S ≈ 117.3187