Write the integral in one variable to find the volume of the solid obtained by rotating the first‐quadrant region bounded by y = 0.5x2 and y = x about the line x = 7

Question

Write the integral in one variable to find the volume of the solid obtained by rotating the first‐quadrant region bounded by y = 0.5x² and y = x about the line x = 7

Answer 1

Let find intersection: 0.5x² =x; x(0,5x - 1) = 0; x = 0 or x = 2

Using Shell method we have: V = 2π∫₀²(7 - x)(x - 0.5x²)dx = 2π∫₀²(0.5x³- 4.5x² + 7x)dx = 2π(x⁴/8 - 1.5x³+ 7x²/2)₀² = 2π(2 - 12 + 14) = 8π