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Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y=8x², x=1, y=0 about the x-axis.

User RafaelGP
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Final answer:

To find the volume of the solid obtained by rotating the region bounded by y=8x², x=1, y=0 around the x-axis, apply the disk method to perform an integral from x=0 to x=1 and calculate the volume, which is 12.8π cubic units.

Step-by-step explanation:

To find the volume of a solid obtained by rotating a curve about an axis, we typically use the disk or washer method. In this problem, we are rotating the region bounded by the parabola y = 8x², vertical line x = 1, and y = 0 around the x-axis to form a solid of revolution.

The volume V of the solid can be found by using the formula for the volume of a disk with radius r(x) and thickness dx, which is:

V = ∫ r(x)² π dx, where r(x) is the function representing the radius of the disk (y = 8x²) and the integral is taken from x = 0 to x = 1.

Plugging in the function we get:

V = π ∫01 (8x²)² dx
= π ∫01 64x⁴ dx
= 64π ∫01 x⁴ dx
= 64π [\(1/5) x⁵]∫01
= 64π [\(1/5) (1)⁵ - (1/5) (0)⁵]
= 64π [\(1/5)]
= 12.8π

The volume of the solid obtained is 12.8π cubic units.

User Darryl Young
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8.1k points
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