THE SHELL METHOD

Question

The region bounded by y=x^2-3x+2 and y=0 is rotated about the y-axis. Find the volume of the resulting solid by any method.

Answer 1

USIND THE CYLINDRICAL SHELL METHOD

V= ∫₁² 2πx [ 0 - ( x²-3x +2)]dx = π / 2

USING THE WASHER METHOD

In this case you need the outer and the inner radius of revolution

by solving the equation x² - 3x +2 -y =0 for x

X_OUTER = 3/2 + 1/2√(1+4x)

X_INNER = 3/2 - 1/2√(1+4x)

Then V= ∫_-1/4⁰ π [ {3/2 + 1/2√(1+4x)}² - {3/2 - 1/2√(1+4x)}² }dy = π/2