Problem 40 - 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. $$ \text{Force} \quad \text{Interval} $$ $$ F(x)=\frac{e^{x^{2}}-1}{100} \quad 0 \leq x \leq 4 $$

Step-by-Step Solution

Answer

The exact answer can be determined using numerical approximation or by using a calculator with integration capabilities. The integral \(\int_{0}^{4} \frac{e^{x^2}-1}{100} \, dx\) represents the total work done by the hydraulic press in foot-pounds when it moves from 0 to 4 feet.

Step 1: Preparing the Integral

The formula for calculating work is given by the integral: \(\text{Work} = \int_{a}^{b} F(x) \, dx\). In this case, function \(F(x)\) is \(\frac{e^{x^2}-1}{100}\) and the interval values of \(x\) are from 0 to 4, as we shall use these values to calculate the definite integral. So, the integral becomes \(\int_{0}^{4} \frac{e^{x^2}-1}{100} \, dx\).

Step 2: Computing the Integral

Compute the integral \(\int_{0}^{4} \frac{e^{x^2}-1}{100} \, dx\) using a numerical integration method or a calculator with integration capabilities.

Step 3: Interpretation of the Result

The result that we obtain after integrating can be interpreted as the total work done by the hydraulic press in moving from 0 to 4 feet. The unit of measurement for work done will be foot-pounds since the force is in pounds and the distance in feet.

Key Concepts

Integration

Integration is a fundamental concept in calculus that helps us find quantities like areas, volumes, and work from known rates of change. It is the reverse process of differentiation. Basically, if we know how something is changing, integration allows us to find the total amount of change over a given interval. When we perform integration, we bring together or "sum up" slices under a curve, which visually represent total accumulation. This is crucial in many areas of science and engineering where determining the cumulative effect of a force, for example, is necessary. Integration can be approached symbolically or numerically, and each method has its own use cases.

Definite Integral

A definite integral gives the total accumulation of a quantity, such as area under a curve, between two specific points on the x-axis. It is written in the form \[\int_{a}^{b} F(x) \, dx\]where \(a\) and \(b\) are the boundaries, or limits, of integration. For the definite integral, unlike indefinite integrals, we calculate a specific numeric value. This value represents the net change caused by a varying rate (given by the function \(F(x)\)) over the interval from \(a\) to \(b\). Definite integrals have many applications, from calculating the work done by a force to finding the net growth in population or the total distance traveled by an object. The integration done in the hydraulic press example is a classic illustration of how definite integrals translate a continuous force over a distance into measurable work.

Numerical Integration

Sometimes integrating a function exactly is very difficult or impossible to do by symbolic means, especially when dealing with complex functions like \( \frac{e^{x^2} - 1}{100} \). This is where numerical integration comes into play. It is an approximation technique that computes the value of an integral using discrete sums rather than continuous functions. Methods such as the trapezoidal rule, Simpson's rule, or using a graphing calculator's integration capabilities help achieve this.

• Trapezoidal Rule: Breaks down the area under the curve into trapezoids and sums the area of these for an approximation.
• Simpson's Rule: Uses parabolic segments to more accurately estimate the area under a curve.

Numerical integration methods are vitally important in real-world applications where precision in extremely complex systems is needed quickly and without extensive calculations.

Work and Energy in Calculus

In calculus, the concept of work is closely tied to the idea of energy. Work is defined as a force causing an object to move a certain distance and is mathematically represented via definite integrals. The formula \( \text{Work} = \int_{a}^{b} F(x) \, dx \) connects these ideas by recognizing that the total work done is the accumulation of force applied over a distance.In the context of physics and engineering, work involves energy transfer. For example, when a hydraulic press applies a force to move an object over a certain distance, it performs work and expends energy in the process. By integrating the force function \( F(x) \) over the distance traveled, we can calculate the exact amount of work done. This measurement, usually expressed in foot-pounds or joules, depends on both the intensity of the force and the distance over which it acts.