Final Answer:
The area between y=x² and y=x⁵ when rotated about the x-axis is:
C) 4π
Step-by-step explanation:
To find the area between the curves y=x² and y=x⁵ when rotated about the x-axis, we can set up the definite integral using the disk method. The formula for the volume generated by rotating the region between the curves about the x-axis is given by ∫[a, b] π(f(x)² - g(x)²)dx, where a and b are the bounds of integration.
In this case, the region is between y=x² and y=x⁵. To find the bounds of integration, we set the two functions equal to each other: x² = x⁵. Solving for x, we get x=0 and x=1 as the bounds. The integral becomes ∫[0, 1] π(x⁵ - x⁴)dx. Evaluating this integral yields the volume of the solid of revolution. Simplifying the expression, we get ∫[0, 1] πx⁴(x - 1)dx. The result of this integral is 4π/5. Therefore, the correct answer is C) 4π.
Understanding the geometric interpretation of integrals is crucial for solving problems involving volumes of revolution. In this case, the disk method is employed to find the volume of the solid formed by rotating the region between the curves about the x-axis. The integral captures the contribution of each infinitesimally thin disk to the total volume, resulting in the correct answer of C) 4π.