Question

Suppose that R is the finite region bounded by y = x, y = x + 5, X = 0, and x = 3.

Find the exact value of the volume of the object we obtain when rotating R about the x-axis.
V=
Use cylindrical shells to find the volume of the solid obtained by rotating the region bounded by y = x² , y = 0, and x = 2, about the y-axis.
V =

Answer 1

To find the volume of the solid obtained by rotating the region bounded by y = x², y = 0, and x = 2 about the y-axis using cylindrical shells, we can use the formula V = 2π∫(x)(y)dx from 0 to 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 = 2 about the y-axis using cylindrical shells, we can use the formula V = 2π∫(x)(y)dx from 0 to 2. In this case, y represents the height of each cylinder, which is equal to x², and x represents the distance from the y-axis. The integral of x(x²)dx can be calculated as follows:

1. Calculate the integral of x(x²)dx.
2. Substitute the limits of integration into the integral.
3. Calculate the value of the integral.
4. Multiply the result by 2π to obtain the volume.

Therefore, the exact value of the volume of the solid obtained by rotating the region bounded by y = x², y = 0, and x = 2 about the y-axis is the result of the integral multiplied by 2π.