Problem 42 - Thomas Calculus

Question

In Exercises \(37-42,\) write a differential formula that estimates the given change in volume or surface area. The change in the lateral surface area \(S=2 \pi r h\) of a right circular cylinder when the height changes from \(h_{0}\) to \(h_{0}+d h\) and the radius does not change

Step-by-Step Solution

Answer

The change in surface area is \( dS = 2 \pi r \cdot dh \).

Step 1: Identify Known Quantities

We know that the lateral surface area of a cylinder is given by \( S = 2 \pi r h \). In this problem, the radius \( r \) remains constant, and the height \( h \) changes from \( h_0 \) to \( h_0 + dh \). We need to determine how this change in height affects the surface area \( S \).

Step 2: Differentiate the Surface Area Formula

To estimate the change in lateral surface area, differentiate the formula \( S = 2 \pi r h \) with respect to \( h \). This results in \( \frac{dS}{dh} = 2 \pi r \). This derivative represents the sensitivity of the surface area to changes in height.

Step 3: Calculate the Change in Surface Area

Now that we have the derivative \( \frac{dS}{dh} = 2 \pi r \), we can calculate the change in the surface area by multiplying the derivative by the change in height \( dh \). Thus, the change in surface area \( dS = 2 \pi r \cdot dh \).

Key Concepts

cylinder surface area

The lateral surface area of a right circular cylinder is a crucial concept in geometry, particularly when calculating areas related to cylindrical shapes.
A right circular cylinder has two main components: the circular bases and the curved lateral surface between them.
In the exercise provided, we focus only on the lateral surface area, calculated by the formula:

• \( S = 2 \pi rh \)

Here, \( r \) is the radius of the circular base, and \( h \) is the height of the cylinder.
This formula arises because the lateral surface of a cylinder can be "unwrapped" into a rectangle, where the height remains the same, \( h \), and the base equals the circumference of the circle, which is \( 2\pi r \).
Understanding how changing one dimension (like height, \( h \)) affects the surface area is key in many practical applications, from engineering to everyday objects like soup cans.

differentiation formula

Differentiation is a fundamental tool in calculus, used to find how a function changes as its input changes.
Essentially, it allows us to determine the rate at which one quantity changes relative to another.
In our exercise, we use differentiation to determine how the lateral surface area of a cylinder responds to changes in its height.

• Given the formula for lateral surface area \( S = 2 \pi rh \), we differentiate with respect to \( h \)

This gives us the derivative \( \frac{dS}{dh} = 2 \pi r \).
The process of differentiation here captures how sensitive the surface area is to changes in the height.
What's important about this derivative, \( 2\pi r \), is that it represents a constant, because the radius \( r \) does not change in this scenario.
Therefore, any change in height directly and linearly affects the surface area.

rate of change

The concept of "rate of change" in calculus describes how one quantity varies with respect to another.
In our previous calculation, we found the rate of change of the cylinder's lateral surface area due to a change in height.
This is expressed by the derivative \( \frac{dS}{dh} = 2 \pi r \).

• The derivative tells us that for every small increase \( dh \) in height, the surface area \( S \) increases by \( 2\pi r \times dh \).

Think about this as a way of measuring the effect of tightening or loosening around the cylinder, only in height direction.
The value \( 2\pi r \) remains constant, showing that any alteration in height has a predictable and precise impact on the surface area.
Rate of change is a powerful concept as it allows for approximations and precise estimates of how quantities interact, especially in real-world scenarios where small changes can occur.