Volume of revolution about the x-axis generated between curves

Volume of Revolution about the x-axis

The volume of revolution around the x-axis is the volume of the solid body that is obtained when you revolve a planar region around the x-axis.
This volume is typically found by using the method of integration.

The general formula for the volume formed by revolving a curve about the x-axis is given by V = π ∫[a, b] [y(x)]² dx.
This tells us that we need to integrate the square of the function, multiplied by π, over the given interval [a, b].
In this formula, a and b are the limits of the region along the x-axis, y(x) is the function defining the curve, and dx indicates that it’s an integral with respect to x.

First, set up the integral using the formula mentioned above. That is, identify the limits a and b, and the function y(x).
Next, compute the integral. This may require using specific integration techniques, like integration by parts, substitution, or partial fractions.
After performing the integration, multiply the result by π to get the volume.
Don’t forget to interpret the results in the context of the problem.

The method of calculating the volume of revolution about the x-axis involves the use of calculus, and specifically the operation of integration.
Sometimes, finding the integral can be quite complex and may involve the use of more advanced integration techniques. It is, therefore, important to brush up your integration techniques.
It is also essential to know how to correctly set up the limits of integration. These limits will be the end points of the region in the x-axis being revolved. -Attention to detail: Make sure to square the function y(x) within the integral, as the formula calls for y(x)².
Consistently practise previous exercises to enhance understanding and speed in solving problems related to volumes of revolution.