Question

Let r be the region bounded by y = x² and y = 0. Use the shell method to find the volume of the solid generated when r is revolved about the line y = 0.

Answer 1

To find the volume of the solid generated by revolving the region between y = x² and y = 0 around the y-axis, use the shell method by integrating the volume of cylindrical shells from 0 to infinity.

Step-by-step explanation:

To calculate the volume of the solid generated by revolving the region bounded by y = x² and y = 0 about the line y = 0, we will use the shell method. The region r is bounded by the parabola y = x² and the x-axis (y = 0). To revolve this region around the y-axis, we consider cylindrical shells that have a height of x², a circumference of 2πx, and a thickness of dx.

The volume dV of a thin shell is given by the product of the shell's height, circumference, and thickness, thus dV = (height)(circumference)(thickness) = (x²)(2πx)(dx) = 2πx³dx. To find the total volume, we integrate this expression from x = 0 to the x-value where the parabola intersects the x-axis. Since the curve y = x² intersects the x-axis at x = 0 and y = 0, we integrate from 0 to ∞. The integral of the volume is calculated as ∫ 2πx³dx from 0 to ∞.