Find the volume of the solid obtained by rotating the region bounded by the curves y=x^2 & y=1 about y=3

Answer 1

Using the washer method:

The line y = 1 intersects the graph of y = x² at x = 1 and at x = -1.

The region R bounded by y = 1 and y = x² is symmetric with respect to the y-axis, so we can integrate from 0 to 1 and then double the result.

At x, take a thin vertical slice of the region with width dx and rotate it about the line y = 3.

The volume of the resulting washer is π[(outer redius)² - (inner radius)²]dx =

π[ (3 - x²)² - (3 - 1)² ]dx = π[9 - 6x² + x⁴ - 4]dx = π[5 - 6x² + x⁴]dx

Volume of solid = 2π∫_{(0 to 1)} (5 - 6x² + x⁴)dx = 2π(5x - 2x³ + (1/5)x⁵)_{(0 to 1)} = 32π/5

Answer 2

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