Volume generated by the revolution of the area bounded by x axis, the catenary y=ccoshx/c, and the coordinates x= -c about the axis of x

To find the volume generated by the revolution of the area bounded by the x-axis, the catenary y=ccoshx/c, and the coordinates x= -c about the x-axis, we can use the method of cylindrical shells.

Step 1: Determine the limits of integration

The given catenary equation is y = c*cosh(x/c), where c is a positive constant. To find the limits of integration, we need to determine the x-values where the catenary intersects the x-axis.

Since the catenary is symmetric about the y-axis, we can focus on the positive x-values. Setting y = 0 in the catenary equation, we have:

0 = c*cosh(x/c)
cosh(x/c) = 0

Since cosh(x/c) is never equal to zero, there are no x-values where the catenary intersects the x-axis. Therefore, the limits of integration for x are from -∞ to ∞.

Step 2: Determine the radius of the cylindrical shell

The radius of each cylindrical shell is equal to the x-coordinate of the catenary at a given value of x. In this case, the radius is x.

Step 3: Determine the height of the cylindrical shell

The height of each cylindrical shell is equal to the difference in y-values between the catenary and the x-axis at a given value of x. In this case, the height is y = ccosh(x/c).

Step 4: Set up the integral

The volume of each cylindrical shell is given by the formula: dV = 2πrh dx, where r is the radius and h is the height.

Therefore, the integral for the volume can be set up as:

V = ∫(from -∞ to ∞) 2πx(ccosh(x/c)) dx

Step 5: Evaluate the integral

To evaluate the integral, we can use integration techniques such as integration by parts or substitution. The resulting integral might be complex and require advanced methods to solve.

Step 6: Simplify the integral and compute the volume

After evaluating the integral, we can simplify the expression and compute the volume using appropriate numerical methods or techniques.

Step 7: Final answer

The final answer will be the value of the volume computed from the integral.

By following these steps, we can find the volume generated by the revolution of the area bounded by the x-axis, the catenary y=ccoshx/c, and the coordinates x= -c about the axis of x.