Question

find the volume using the disc method
f(x)x^2 between x=-2 and x=2

Answer 1

Assuming the region bounded by y = x^2, the x-axis, and the vertical lines x = −2 and x = 2 is rotated about the x-axis, the radius of each disc is y = x^2. The volume is

V = π ∫_{−2}^{2} (radius)^2 dx = π ∫_{−2}^{2} (x^2)^2 dx = π ∫_{−2}^{2} x^4 dx.

Use symmetry or compute directly:
V = 2π ∫_{0}^{2} x^4 dx = 2π [x^5/5]_{0}^{2} = 2π (32/5) = 64π/5.

So the volume is 64π/5 cubic units.