Question

What is the volume of the solid generated when the region bounded by the graph of x = √(y-2) and the lines x=0 and y=5 revolved about the y axis?

Answer 1

To find the volume of the solid generated, we can use the method of cylindrical shells. This involves integrating the circumference of each shell multiplied by its height. Using the formula, the volume is calculated as π(39/2) cubic units.

Step-by-step explanation:

To find the volume of the solid generated when the region bounded by the graph of x = √(y-2) and the lines x=0 and y=5 is revolved about the y-axis, we can use the method of cylindrical shells. This method involves integrating the circumference of each shell multiplied by its height.

First, we need to express the equation x = √(y-2) in terms of x, which gives us y = x^2 + 2. The range of integration will be from x=0 to x=√3 (the point where the curve intersects y=5).

Using the formula for the volume of a cylindrical shell, V = 2π ∫(radius)(height)dx, we can calculate the volume as follows:

1. Calculate the radius of each shell: r = x
2. Calculate the height of each shell: h = y - 0 = y
3. Integrate the product of the radius and height over the given range: V = 2π ∫(x)(y) dx = 2π ∫(x)(x^2 + 2) dx
4. Simplify the integral and evaluate it: V = 2π ∫(x^3 + 2x) dx = 2π(x^4/4 + x^2) from 0 to √3
5. Substitute the limits of integration and evaluate the integral: V = 2π[(√3)^4/4 + (√3)^2 - 0] = 2π(27/4 + 3) = 2π(27/4 + 12/4) = 2π(39/4)
6. Simplify the result: V = π(39/2)

Therefore, the volume of the solid generated is π(39/2) cubic units.

Answer 2

The volume of the solid generated when the region bounded by the graph of x = √(y-2) and the lines x=0 and y=5 revolved about the y-axis is 40π/3 cubic units.

Step-by-step explanation:

To find the volume of the solid formed by revolving the region around the y-axis, we can use the disk method. The given region is bounded by x = √(y-2), x = 0, and y = 5. First, we need to find the limits of integration. Setting x = 0 gives us the lower limit, and solving √(y-2) = 0 provides the upper limit. Thus, the limits of integration are from y = 2 to y = 5.

Now, the formula for the volume using the disk method is given by:

V = π ∫[a to b] [f(y)]² dy

Substituting the given function x = √(y-2) into the formula, we get:

V = π ∫[2 to 5] [√(y-2)]² dy

Simplifying further, we have:

V = π ∫[2 to 5] (y-2) dy

Evaluating this integral yields the final result of 40π/3 cubic units for the volume of the solid formed by revolving the given region around the y-axis.