Area bounded and Washers Volume

Question

The area bounded by y= 4-x² and the x-axis is rotated about the line y= -3.

a.) Graph the region and determine points at which the graphs intersect.

b.) Use washers to compute the volume generated.

Answer 1

The region to be rotated is bounded above by the graph of y = 4 - x² and below by the x-axis.

The graph of y = 4 - x² intersects the x-axis when x = ±2.

Volume of a typical washer generated by rotating a vertical cross section of the region about the line y = -3 is

π(3 + (4-x²))²dx - π(3)²dx = π(7 - x²)²dx - 9πdx

= π(49 - 14x² + x⁴ - 9)dx

= π(40 - 14x² + x⁴)dx

Volume of solid = π∫_{(from -2 to 2)} [40 - 14x² + x⁴]dx

= 2π∫_{(from 0 to 2)} [40 - 14x² + x⁴]dx (By symmetry)

= 2π[40x - (14/3)x³ + (1/5)x⁵]_{(from 0 to 2)}

= 2π(80 - (112/3) + (32/5)) = 1472π / 15