Question

find the volume using the disc method
f(x)= 5x - 2x^2 between x=-1 and x=3

Answer 1

Assuming the region between the curve and the x-axis from x = -1 to x = 3 is revolved about the x-axis, the radius of the disc is |f(x)|, so

V = π ∫_{-1}^{3} [f(x)]^2 dx = π ∫_{-1}^{3} (5x - 2x^2)^2 dx.

Expand and integrate:
(5x - 2x^2)^2 = 4x^4 - 20x^3 + 25x^2,
antiderivative = (4/5)x^5 - 5x^4 + (25/3)x^3.

Evaluate from -1 to 3:
V = π [ (4/5)(3^5) - 5(3^4) + (25/3)(3^3) - ( (4/5)(-1)^5 - 5(-1)^4 + (25/3)(-1)^3 ) ]
= π * (428/15) = (428π)/15 ≈ 89.59.

So V = 428π/15 (≈ 89.59).