Final answer:
To find the volume of the solid obtained by rotating the region bounded by y=x², y=0, and x=1, we can use cylindrical shells. The volume is given by the integral of 2πx³ from 0 to 1, which evaluates to π/2.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by y=x², y=0, and x=1, we can use cylindrical shells. Cylindrical shells are formed by integrating the product of the circumference of the shell and the height of the shell.
First, we need to determine the limits of integration. The region is bounded by y=x², y=0, and x=1. Since the region is symmetric with respect to the y-axis, we can integrate from 0 to 1.
The radius of each cylindrical shell is x, and the height is given by y=x². The circumference of the shell is 2πx. Integrating over the region, the volume is:
V = ∫(2πx) * (x²) dx, where the limits of integration are from 0 to 1.
Simplifying the integral, we have:
V = ∫2πx³ dx, where the limits of integration are from 0 to 1.
Integrating, we get:
V = π/2