Find the volume of the solid obtained by rotating the region enclosed by the graphs about the axis. 𝑓(𝑥)=9/𝑥^2, 𝑔(𝑥)=10−𝑥^2, 𝑥≥0, about 𝑥=−1

Answer 1

Positive x-coordinate of intersection points: 10 - x² = 9/x², x⁴ - 10x² + 9 = 0; x² = 1, x = 1 and x² = 9, x = 3.

By Shell method volume V =2π∫₁³(x + 1)(10 - x² - 9/x²)dx =2π∫₁³(10x - x³ - 9/x + 10 - x² - 9/x²)dx =

(5x² - x⁴/4 - 9lnx + 10x - x³/3 + 9/x)₁³ = 2π[(45 - 81/4 - 9ln3 + 30 - 9 +3) - (5 - 1/4 -9ln1 + 10 - 1/3 + 9)] ≈ 97.05