Question

Use cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y=x², y=0, x=4; about the y-axis

Answer 1

To find the volume of the solid obtained by rotating the region bounded by the given curves about the y-axis using cylindrical shells, we need to integrate the formula for the volume of a cylindrical shell. The volume of each cylindrical shell is given by 2πrhΔx, where h is the height of the shell and Δx is the thickness of the shell. We can set up the integral to find the total volume by integrating 2πx(x²)dx from x = 0 to x = 4.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by the curves y = x², y = 0, x = 4 about the y-axis using cylindrical shells, we need to integrate the formula for the volume of a cylindrical shell. The volume of each cylindrical shell is given by 2πrhΔx, where h is the height of the shell and Δx is the thickness of the shell.

We can set up the integral to find the total volume by integrating 2πx(x²)dx from x = 0 to x = 4. This represents the sum of the volumes of all the cylindrical shells as x ranges from 0 to 4. After integrating, the integral becomes 2π∫[0 to 4]x³dx, which can be evaluated to find the volume of the solid obtained.

Let's calculate the volume using this formula:

1. Set up the integral: ∫[0 to 4]2πx(x²)dx
2. Integrate: 2π∫[0 to 4]x³dx
3. Evaluate the integral: 2π[x⁴/4] from 0 to 4
4. Calculate the final volume: V = 2π(4⁴/4) - 2π(0⁴/4) = 32π

The volume of the solid obtained by rotating the region bounded by the given curves about the y-axis is 32π cubic units.