Question

Volume The change in the volume \(V=\pi r^{2} h\) of a right circular cylinder when the radius changes from \(a\) to \(a+d r\) and the height does not change

Short answer

The change in volume when the radius of the right circular cylinder changes from \(a\) to \(a+dr\) is \(2\pi a h dr\).

Step-by-step solution

Understanding the volume formula of a cylinder

The given formula \(V=\pi r^{2} h\) tells us that the volume of a cylinder is equal to the area of its base (which is a circle with radius \(r\), so its area is \(\pi r^{2}\)) multiplied by its height \(h\).

Differentiating the volume formula

We want to find the change in volume with respect to a small change in radius. This requires differentiating the volume formula with respect to \(r\). Using the power rule that the derivative of \(r^{2}\) is \(2r\), we get the derivative of the volume formula as \(dV = 2\pi r h dr\).

Substituting the values of radius and change in radius

To find the actual change in volume when the radius changes from \(a\) to \(a+dr\), we substitute \(r=a\) and \(dr=dr\) into the equation from Step 2: \(dV = 2\pi a h dr\).

Key concepts

Differential Calculus

Differential calculus is a subfield of calculus concerned with the notion of a derivative, which represents an instantaneous rate of change. This is particularly useful when analyzing how a small change in one quantity leads to a change in another. In geometry and physics, we often use differential calculus to determine how objects alter their shape or velocity.

With the example of a cylinder's volume changing, differential calculus allows us to calculate the effect of a slight alteration in one dimension, say the radius, while keeping others constant. By understanding the principles of differential calculus, we can predict and understand how real-world systems behave under small perturbations.

Volume of a Cylinder

The volume of a cylinder is a measurement that tells us how much three-dimensional space it occupies. The formula for the volume is given by the product of the base area and the height of the cylinder, expressed as \( V = \text{pi} r^{2} h \).

Where \( \text{pi} \) is a mathematical constant (approximately 3.14159), \( r \) is the radius of the cylinder's circular base, and \( h \) is its height. Understanding this formula is crucial for solving problems related to the storage, displacement, or manufacturing of cylindrical objects.

Power Rule

The power rule is a basic and extremely useful principle in differential calculus for finding derivatives. It states that if you have a function of the form \( f(x) = x^n \), where \( n \) is any real number, then the derivative of that function with respect to \( x \) is \( f'(x) = nx^{n-1} \).

For instance, the derivative of \( r^{2} \) with respect to \( r \) is \( 2r \), since here \( n = 2 \). The power rule simplifies the process of differentiation greatly, making it easier to solve problems involving rates of change in variables raised to a power, such as the radius in the volume of a cylinder.

Derivatives in Geometry

Derivatives play a significant role in geometry, especially when determining how changes in dimensions affect various geometric properties such as volume and surface area. In the context of a cylinder, if we want to know how a small change in the radius affects the overall volume, we would take the derivative of the volume formula with respect to the radius.

This application of derivatives helps architects, engineers, and designers to calculate necessary materials, understand structural stability, and optimize shapes for different functions. By mastering derivatives in the realm of geometry, students can gain valuable insights into the relationships between different geometric dimensions.