Problem 42 - Calculus of a Single Variable

Question

Hydraulic Press In Exercises \(39-42,\) use the integration capabilities of a graphing utility to approximate the work done by a press in a manufacturing process. A model for the variable force \(F\) (in pounds) and the distance \(x\) (in feet) the press moves is given. $$ \text{Force} \quad \text{Interval} $$ $$ F(x)=1000 \sinh x \quad 0 \leq x \leq 2 $$

Step-by-Step Solution

Answer

The work done by the hydraulic press is approximately 3523.52 pounds-feet.

Step 1: Understand the Force Function F(x)

The force exerted by the hydraulic press is given by \(F(x) = 1000 \sinh x\). Notice that as \(x\) changes from 0 to 2, the force \(F(x)\) behaves differently. This means the force is variable.

Step 2: Setup the Integration

The work done by a variable force over a distance is calculated as the integral of the force over the distance. In this case, the work done \(W\) is the integral of the force function \(F(x) = 1000 \sinh x\) from 0 to 2. Hence, \(W = \int_{0}^{2} 1000 \sinh x dx\).

Step 3: Compute the Integration

Utilizing a graphing utility or integral calculator, evaluate the integral \(\int_{0}^{2} 1000 \sinh x dx\). Which comes out to be approximately 3523.52 pounds-feet.

Key Concepts

Integration

Integration is a fundamental concept in calculus that can be thought of as the process of finding the whole from the parts. It allows us to calculate quantities like area, volume, and in this case, the work done when a force is applied over a distance. The work done by a force can be represented by the area under the force-distance curve on a graph.

When dealing with varying forces, integration becomes even more crucial as it helps account for the change in force over the distance moved. In the context of the hydraulic press exercise, integration is used to determine the total work done when the force exerted by the press is not constant but varies with the distance pressed, modeled by the function \( F(x) = 1000 \sinh x \).

To calculate the total work \(W\), the integration of \(F(x)\) from the starting position \(x=0\) to the end position \(x=2\) feet needs to be performed:\[ W = \int_{0}^{2} 1000 \sinh x \,dx \].
The integration process accounts for the continuous variation of force and ensures a precise calculation of work, in contrast to simple multiplication that would only be accurate for a constant force.

Variable Force

A variable force is a force that changes in magnitude and/or direction as the point of application moves. In many real-world applications, like hydraulics, the force exerted is not constant and depends on various factors, such as distance, pressure, or time.

In the hydraulic press example, the force applied by the press is represented by the function \( F(x) = 1000 \sinh x \), where \(x\) is the distance the press moves. The hyperbolic sine function, \(\sinh(x)\), describes how the force magnifies as the distance increases, which is typical in some manufacturing processes utilizing presses.

Understanding variable forces is essential in engineering and physics because it enables accurate predictions of the energy requirement, work done, and ultimately the system's performance. When dealing with variable forces, the concept of integration is often employed to find the total effect over a range, offering a more nuanced and complete analysis than assuming a constant force.

Hyperbolic Functions

Hyperbolic functions, such as the hyperbolic sine function \(\sinh\), are analogs to the familiar trigonometric functions but are based on hyperbolas instead of circles. These functions are useful in various areas, including engineering, physics, and mathematics, particularly when dealing with exponential growth or decay phenomena.

In our exercise, the force exerted by the hydraulic press is described using the hyperbolic sine function: \(F(x) = 1000 \sinh x\). This function represents an increasing force that grows more quickly than a linear function would as \(x\) increases. The hyperbolic sine function can be defined as:\[ \sinh x = \frac{e^x - e^{-x}}{2} \],where \(e\) is Euler's number, an important constant in mathematics. This definition illustrates that \(\sinh x\) combines growth and decay terms, which results in a smooth and continuously increasing function.

Hyperbolic functions often arise in systems that display 'coshed' or 'sinhed' behavior, meaning systems where phenomena grow or decay in a manner similar to the shape of a hyperbolic curve. Their applications include modeling suspension bridges, electrical engineering, and as seen in our example, understanding the force applied by industrial machinery.