Question

Hydraulic Press In Exercises 25 and \(26,\) 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. $$ F(x)=\frac{e^{x^{3}}-1}{100} \quad 0 \leq x \leq 4 $$

Short answer

The work done by the press is given by the definite integral \( \int_{0}^{4} F(x) dx \), which we need to estimate using a graphing utility due to the complexity of the integral.

Step-by-step solution

Understand the Problem and the Given Function

We are given the function \( F(x)=\frac{e^{x^{3}}-1}{100} \), which represents the variable force applied by the press over the distance \( x \), between 0 and 4 feet. The work done by a force is found by taking the integral of the force function over its range of application.

Set Up the Integral for Work

Work is the integral of force over the distance over which it is applied. So, we need to integrate the function \( F(x) \) from 0 to 4. Thus, the work \( W \) done by the press is given by \( W=\int_{0}^{4} F(x) dx = \int_{0}^{4} \frac{e^{x^{3}}-1}{100} dx \).

Use a Graphing Utility to Approximate the Integral

The integral we obtain in step 2 is not a standard integral and it's difficult to solve analytically. Hence, we use a graphing utility or a calculator with integral capabilities to estimate this integral. Plug in the function and the limits of integration into the utility to get the numerical approximation.

Key concepts

Integration in Calculus

Integration in calculus is a fundamental concept that deals with the process of finding the total amount of some quantity given its rate of change. It's essentially the reverse process of differentiation, and it can be thought of as 'adding up' infinitesimal 'slices' to find a cumulative total. When we integrate a function over a certain interval, we are finding the area under the curve of that function within that interval.

For instance, if you are given a graph of speed versus time, and you want to determine the distance traveled during a certain time period, you would integrate the speed function over that time interval. This concept is what we apply when we calculate the work done by the press in the given scenario. We interpret the function F(x) as a force applied over a distance, and so the integral of that function from 0 to 4 feet represents the total work done.

To set up an integral in a calculus problem, one must identify the variable representing the rate of change and the interval over which to integrate. The integral can either be evaluated analytically, using anti-derivatives, or numerically, which is often necessary for more complex functions like in our exercise.

Variable Force Integration

In the context of work and physics, when a force varies as it is applied over a distance, we use variable force integration to determine the total work done. It differs from the scenario where the force is constant because the amount of work done changes at different points along the distance over which the force is applied.

Mathematically, the work W done by a variable force F(x) from point a to point b is calculated using the integral W=\text{\(\int_{a}^{b} F(x) dx\)}. In the given exercise, the force applied by the hydraulic press doesn't remain constant but changes according to the function F(x) = \text{\(\frac{e^{x^3}-1}{100}\)}.

Therefore, to find the work done by the press, we do not simply multiply a constant force by distance. Instead, we integrate the given force function F(x) over the distance from 0 to 4 feet, applying calculus to account for how the force changes at each infinitesimal increment of distance.

Graphing Utility for Integration

A graphing utility can be incredibly helpful when dealing with complex integrals that do not have straightforward anti-derivatives. As the name suggests, it is a tool—often a feature of advanced scientific calculators or computer software—that allows us to plot graphs and perform various functions like finding roots, maximums, minimums, and, crucially for our purposes, numerical integration.

When an integral is too difficult to solve analytically or if we just need a quick estimation, we can input the function into a graphing utility, along with the limits of integration, and it will use numerical methods to approximate the value of the integral. These methods, such as the trapezoidal rule or Simpson's rule, work by approximating the area under the curve with shapes (like trapezoids or parabolas) whose areas we can easily calculate.

For the hydraulic press problem, using a graphing utility saves us the trouble of having to solve the complex integral by hand. It allows students to quickly and efficiently find an approximate value for the work done, which is particularly useful for verifying manual calculations or when the precise analytical solution is not necessary.