Question

Find the volume V of the solid obtained by rotating the region bounded by the curves y = x³, y = 1, x = 2 about the line y = -5.

Answer 1

To find the volume of the solid obtained by rotating the region bounded by the curves y = x³, y = 1, and x = 2 about the line y = -5, you can use the method of cylindrical shells. Integrate the product of the circumference and height of each shell over the range of x values to find the total volume of the solid.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by the curves y = x³, y = 1, and x = 2 about the line y = -5, we can use the method of cylindrical shells.

We integrate the volume of each shell from the range of x values that correspond to the region bounded by the curves. In this case, the range is from x = 1 to x = 2.

1. First, we find the radius of each shell, which is the distance from the line y = -5 to the curve y = x³. Since the curve is below the line, the radius is given by r = -5 - x³.
2. Next, we find the height of each shell, which is the difference between the upper and lower functions. In this case, the height is h = 1 - x³.
3. Then, we calculate the circumference of each shell, which is given by c = 2πr.
4. Finally, we integrate the product of the circumference and height over the range of x values to find the total volume V of the solid. This can be written as V = ∫[(1 - x³)(2π(-5 - x³))]dx.