What is the volume generated when the region surrounded by y = √x, y = 2 and y = 0 is revolved about y – axis?a)32π cubic unitsb)32 / 5 cubic unitsc)5π / 32 cubic unitsd)32π / 5 cubic unitsCorrect answer is option 'D'. Can you explain this answer?

Question

What is the volume generated when the region surrounded by y = &radic;x, y = 2 and y = 0 is revolved about y &ndash; axis?a)32&pi; cubic unitsb)32 / 5 cubic unitsc)5&pi; / 32 cubic unitsd)32&pi; / 5 cubic unitsCorrect answer is option 'D'. Can you explain this answer?

Answer

To find the volume generated when the region surrounded by y = sqrt(x), y = 0, and x = 16 is rotated about the y-axis, we can use the method of cylindrical shells.

First, let's find the limits of integration. The region is bounded by y = sqrt(x), y = 0, and x = 16. To find the limits of integration, we can solve the equation y = sqrt(x) for x:

y = sqrt(x)
x = y^2

So, the limits of integration for x are from 0 to 16. The limits of integration for y are from 0 to the top curve, which is y = sqrt(x).

Next, let's set up the integral to calculate the volume. The formula for the volume of a solid generated by rotating a region about the y-axis using cylindrical shells is:

V = ∫[a, b] 2πx * h(x) * dx

where a and b are the limits of integration, h(x) is the height of the shell at x, and dx is the thickness of the shell.

In this case, the height of the shell at x is given by the difference between the top curve, y = sqrt(x), and the bottom curve, y = 0. So, h(x) = sqrt(x) - 0 = sqrt(x), and dx is the thickness of the shell.

The volume can then be calculated as:

V = ∫[0, 16] 2πx * sqrt(x) * dx

Simplifying the integral:

V = 2π ∫[0, 16] x^(3/2) * dx

Using the power rule for integration:

V = 2π * (2/5) * x^(5/2) | from 0 to 16

Evaluating the integral:

V = 2π * (2/5) * (16^(5/2) - 0^(5/2))
V = 2π * (2/5) * (256 - 0)
V = 2π * (2/5) * 256
V = 2π * (512/5)
V = 1024π/5

So, the volume generated when the region surrounded by y = sqrt(x), y = 0, and x = 16 is rotated about the y-axis is 1024π/5 cubic units.

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