Volume of a Cylinder

To find the volume of a cylinder, we need to know the formula for the volume of a cylinder. The formula is:

V = πr^2h

where V is the volume, π is a mathematical constant approximately equal to 3.142, r is the radius of the circular base, and h is the height of the cylinder.

Equation of the Cylinder

The equation x^2 + y^2 = 2x represents a cylinder in three-dimensional space. This equation can be rewritten in the form (x-1)^2 + y^2 = 1, which is the equation of a circle with center (1, 0) and radius 1.

Bounded by the Planes

The cylinder is bounded by the planes z = 0 and z = x. This means that the cylinder extends from the plane z = 0 up to the plane z = x. The height of the cylinder varies depending on the value of x.

Calculating the Volume

To calculate the volume of the cylinder, we need to integrate the cross-sectional area of the cylinder over the range of x values.

The cross-sectional area of the cylinder is given by A = πr^2, where r is the radius of the circular cross-section. Since the equation of the cylinder is (x-1)^2 + y^2 = 1, the radius r can be calculated as r = √(1 - (x-1)^2).

The range of x values over which the cylinder extends is from 0 to x. Therefore, the volume of the cylinder can be calculated as:

V = ∫[0 to x] A dx

Substituting the value of A, we have:

V = ∫[0 to x] π(√(1 - (x-1)^2))^2 dx

Simplifying the expression, we get:

V = ∫[0 to x] π(1 - (x-1)^2) dx

= ∫[0 to x] π(1 - (x^2 - 2x + 1)) dx

= ∫[0 to x] π(2x - x^2) dx

= π∫[0 to x] (2x - x^2) dx

Integrating the expression, we get:

V = π(x^2 - (x^3)/3)

Final Answer

We need to evaluate the expression π(x^2 - (x^3)/3) over the range of x values from 0 to x. Substituting x = x, we get:

V = π(x^2 - (x^3)/3)

For the given question, the range of x values is not provided. Therefore, we cannot calculate the exact volume. But the correct answer given is approximately 3.142, which is the value of π.