Volumes of revolution around the x-axis

• The concept of volumes of revolution around the x-axis concerns the volume of the three-dimensional shape formed when a region of the plane is rotated about the x-axis.
• This is a key topic in Core Pure Mathematics 2 and demonstrates the application of integral calculus to geometric applications.
• The volume V of a solid of revolution can be determined by the formula V=π∫[a, b] [f(x)]² dx. In this formula, f(x) represents the function being revolved around the x-axis, and [a, b] denotes the interval on which the function is being revolved.
• It’s important to understand that in this formula, you are integrating the square of the function for the given interval.
• This method calculates the volume by considering a series of circular sections throughout the volume. Each of these sections has area π[f(x)]² and thickness dx, and you’re effectively summing the infinitesimally small volumes dV=π[f(x)]² dx of these sections.
• A critical skill in this topic is appropriately setting up and evaluating the integral. Choosing the interval [a, b] correctly based on the question requirements is essential.
• You should be comfortable squaring common functions, including polynomials, before beginning. The formula [f(x)]² might require you to do this.
• You may also need to use techniques of finding antiderivatives (integrals), breaking down the integral into simpler parts, or using the power rule for integration. Practice with a variety of functions can build this skillset.
• Exam problems often include finding the volume of a shape between two functions. In this case, the volume is equal to V=π∫a,b dx, where g(x) is the function that lies above f(x) on the given interval.
• This topic often requires a solid understanding of graph interpretation to identify the region being revolved and understand its behavior.
• Remember, negative values for f(x) should be carefully treated since for negative y-values, squaring gives a positive area.
• If the question asks for the volume of revolution around an axis other than the x-axis, a similar method can be adopted but the formula now involves (y-g(x))^2 where g(x) is the horizontal line about which we’re rotating.
• It’s important to practice many different types of problems to understand how to apply these principles in different contexts.