Explanation:
the volume of a rotating solid out of an xy area in a certain interval is pi times the integral of that area squared (because we are emulating adding the volumes of cylinder slices) in that interval along the given axis.
if that area is defined as the difference between 2 curves, then that difference (of the squares) has to be integrated.
the "outer" curve is y = x² + 3 = R, because it has always the larger y values compared to y = 2x + 1 = r.
the volume of an outer cylinder slice is
pi × R² × dx
and of an internal cylinder slice is
pi × r² × dx
therefore, the volume of the difference "washer" cylinder is
pi×R²×dx - pi×r²×dx = pi×(R² - r²)×dx
and now integrating the infinitely thin washers across the interval [0, 2].
R² = (x²+3)² = x⁴ +6x² + 9
r² = (2x+1)² = 4x² + 4x + 1
R² - r² = x⁴ + 2x² - 4x + 8
the integral of (R² - r²)dx :
1/5 x⁵ + 2/3 x³ - 4/2 x² + 8x = 1/5 x⁵ + 2/3 x³ - 2x² + 8x
in the interval of [0, 2] that gives us
1/5 2⁵ + 2/3 2³ - 2×2² + 8×2 - 0 = 32/5 + 16/3 - 8 + 16 =
= 32/5 + 16/3 + 8 = 96/15 + 80/15 + 120/15 = 296/15
and as final step multiplying this by pi :
296pi/15 = 61.99409503... units³ ≈ 62 units³