Final answer:
To find the volume of the solid formed by revolving the region bounded by y = x⁴ and y = x⁷ about the x-axis, the Washer Method is used. The intersection points of the curves are x = 0 and x = 1, creating the bounds of the definite integral. The final computed volume of the solid is 2π/45 cubic units.
Step-by-step explanation:
Evaluating the Definite Integral for Volume of a Solid of Revolution
To find the volume of the solid formed by revolving the region bounded by y = x⁴ and y = x⁷ about the x-axis, we use the Washer Method. This involves computing the volume as the integral of the difference of the squares of the functions being revolved. We take the integral of π(x⁴)² - π(x⁷)² with respect to x over the interval where the two curves intersect.
First, we find the points of intersection by setting x⁴ equal to x⁷, which implies that x = 0 or x = 1. Therefore, the bounds of integration are from 0 to 1. The integral becomes:
∫₀¹ π(x⁴² - x⁷²) dx = π ∫₀¹ (x⁸ - x¹⁴) dx
We then compute the integral, simplifying to find the volume:
π [¹/¹(x⁸) - ¹/¹⁵(x¹⁴)] from 0 to 1 = π [(1/9) - (1/15)]
This results in a final volume of:
π [(1/9) - (1/15)] = π [5/45 - 3/45] = π [2/45]
Thus, the volume of the solid is 2π/45 cubic units.