Applications of Integration

Question

Find the volume of the solid obtained by rotating the region bounded by the given curves about the line x=−8

y=x², x=y²

Answer 1

Limits of integration: x = x⁴; x = 0 or x = 1.

We applying shell method V = 2π∫₀¹(x + 8)(√x - x²)dx = 2π∫₀¹(x^3/2 - x³ + 8x^1/2 - 8x²)dx =2π(2/5x^5/2 - x⁴/4 +

16/3x^3/2 - 8x³/3)₀¹ =2π(2/5 - 1/4 + 16/3 - 8/3) = 169π/30 = 17.6976