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Evaluate the definite integral that represents the volume of the solid formed by revolving the region about the x-axis. The region is bounded by the curves y = x² and y = x⁷?

User Manismku
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Final answer:

The volume of the solid formed by revolving the region bounded by y = x² and y = x⁷ about the x-axis is found using the washer method. The integral is set up and evaluated between the intersection points of the curves, which are x = 0 and x = 1. The definite integral of π∫ [(x²)² - (x⁷)²] dx from 0 to 1 yields the volume.

Step-by-step explanation:

Evaluating the Volume of the Solid of Revolution:

To determine the volume of a solid formed by revolving the region bounded by the curves y = x² and y = x⁷ about the x-axis, we use the washer method. This involves calculating the volume of infinitesimally thin washers or disks and integrating these from the points where the curves intersect. The volume V of the solid is thus given by the integral:

V = π ∫ [R(x)² - r(x)²] dx

where R(x) is the radius of the outer curve and r(x) is the radius of the inner curve when the solid is oriented along the x-axis. The bounds of integration are where the two functions intersect, which can be found by setting x² = x⁷ and solving for x.

To calculate the integral, we first find the points of intersection by solving the algebraic equation x² = x⁷, indicating that x = 0 or x = 1. Subsequently, we set up the integral with R(x) = x² and r(x) = x⁷, obtaining:

V = π ∫_{0}^{1} [(x²)² - (x⁷)²] dx

Then, we evaluate this definite integral to obtain the volume.

User Eiki
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