The Cissoid

The given equation represents a curve known as the cissoid. The cissoid is a plane curve defined by a certain equation. In this case, the equation of the cissoid is y^2(2a - x) = x^3, where a is a given constant.

Revolution of the Cissoid

We are asked to find the volume of the solid generated by the revolution of the cissoid about the line x = 2a. To do this, we need to use the method of cylindrical shells.

Method of Cylindrical Shells

The method of cylindrical shells involves finding the volume of a solid by integrating the volumes of infinitely thin cylindrical shells.

To apply this method, we need to consider an infinitesimally small strip of width dx along the x-axis. This strip will have a corresponding height, which is given by the y-coordinate of the cissoid curve.

Setting up the Integral

To find the volume of the solid, we need to integrate the volumes of all these cylindrical shells from x = 0 to x = 2a.

The volume of an individual cylindrical shell is given by the formula: V = 2πy * h * dx, where y is the y-coordinate of the cissoid curve and h is the height of the cylindrical shell.

To find y in terms of x, we rearrange the equation of the cissoid: y^2(2a - x) = x^3. Solving for y, we get: y = (x^3) / sqrt(2a - x).

Integrating the Volume

Now we can set up the integral to find the volume of the solid:

V = ∫[0 to 2a] (2πy * h * dx) = ∫[0 to 2a] 2π * (x^3) / sqrt(2a - x) * dx.

To evaluate this integral, we can make a substitution: u = 2a - x. This gives us du = -dx, and we can rewrite the integral as:

V = ∫[2a to 0] -2π * (u^3) / sqrt(u) * (-du) = 2π * ∫[0 to 2a] (u^3) / sqrt(u) * du.

Simplifying the integrand, we get: V = 2π * ∫[0 to 2a] u^(7/2) * du.

Integrating this expression, we get: V = 2π * [2/9 * u^(9/2)] [0 to 2a] = 4π * (2/9 * (2a)^(9/2)).

Substituting the Value of a

Given that a = 1, we can substitute this value into the expression for the volume:

V = 4π * (2/9 * (2(1))^(9/2)) = 4π * (2/9 * 2^(9/2)) = 8π/9 * 2^(9/2).

Calculating this value, we get V ≈ 19.7392.

Final Answer