Question

Let the region R be the area enclosed by the function f(x) = sqrt(x), the horizontal line y = 1, and the y-axis. Find the volume of the solid generated

Answer 1

To find the volume of the solid generated by the region R enclosed by the function f(x) = sqrt(x), the horizontal line y = 1, and the y-axis, we can use the method of cylindrical shells. Each shell will be a thin strip of height dx and thickness dy. The volume of each shell is 2πx(f(x)-1)dx, which we integrate over the range x = 0 to x = 1 to find the total volume. The final answer is π/5 cubic units.

Step-by-step explanation:

The region R is the area enclosed by the function f(x) = sqrt(x), the horizontal line y = 1, and the y-axis. To find the volume of the solid generated, we can use the method of cylindrical shells. Each shell will be a thin strip of height dx and thickness dy. The circumference of the shell is 2πx and the height of the shell is f(x) - 1. The volume of each shell is the product of its circumference, height, and thickness, which is 2πx(f(x)-1)dx.

To find the total volume, we integrate the volume of each shell from x=0 to x=1, which gives:

1. ∫[from 0 to 1] 2πx(f(x)-1)dx
2. 2π ∫[from 0 to 1] (x * sqrt(x) - x)dx
3. 2π ∫[from 0 to 1] (x^(3/2) - x)dx
4. 2π [ (2/5)x^(5/2) - (1/2)x^2 ] [from 0 to 1]
5. 2π [(2/5) - (1/2)]
6. 2π/10
7. π/5

Therefore, the volume of the solid generated is π/5 cubic units.