Problem 32 - Calculus and its applications

Question

Find the area under the graph of each function over the given interval. $$ y=2 x+\frac{1}{x^{2}} ; \quad[1,4] $$

Step-by-Step Solution

Answer

The area under the graph is 15.75.

Step 1: Introduction to Definite Integral

To find the area under the graph of the function over the given interval, we need to compute a definite integral from 1 to 4 of the given function.

Step 2: Setting up the Integral

Write the definite integral for the function: \[\int_{1}^{4} \left(2x + \frac{1}{x^2}\right) \, dx\]

Step 3: Integrate Each Term Separately

Integrate each term separately. The integral of the first term is \[\int 2x \, dx = x^2\] The integral of the second term is \[\int \frac{1}{x^2} \, dx = \int x^{-2} \, dx = -x^{-1} = -\frac{1}{x}\]

Step 4: Combine the Integrals

Combine the results of the integrals: \[\int \left(2x + \frac{1}{x^2}\right) \, dx = x^2 - \frac{1}{x}\]

Step 5: Evaluate the Definite Integral

Evaluate the expression from step 4 from 1 to 4: \[(x^2 - \frac{1}{x})\Bigg|_{1}^{4} = \left(4^2 - \frac{1}{4}\right) - \left(1^2 - \frac{1}{1}\right)\] Simplify to find the value: \[(16 - 0.25) - (1 - 1) = 15.75\]

Key Concepts

Calculus Problem Solving

Calculus problem solving often involves a series of logical steps to find solutions, especially when dealing with definite integrals. It's akin to solving a puzzle by identifying the main problem and breaking it into smaller tasks.
In the provided exercise, the main goal is to find the area under a curve, which is essentially the same as solving a definite integral from 1 to 4 for a given function. A definite integral is a crucial concept in calculus used to calculate area, displacement, and various other quantities.
By setting up the integral, calculating each part separately, and then evaluating it within the given limits, the solution becomes more approachable. It all begins by correctly interpreting the function and the interval. Ensuring that the initial setup is correct paves the path for successful calculations later on.
Key steps to remember in problem solving include:

• Understanding and defining the problem clearly.
• Setting up the mathematical model or equation.
• Breaking down the solution process into manageable parts.
• Carefully calculating each part and combining them correctly.
• Evaluating results and interpreting the final solution.

By systematically following these steps, calculus problems become more manageable and less intimidating.

Integration Techniques

Integration techniques are essential tools for solving calculus problems, particularly when integrating functions to find definite integrals. Understanding these methods can ease the integration process considerably.
In this exercise, we employed basic integration techniques to handle a function composed of two elements: a linear term and a rational term.
The first term, \(2x\), is straightforward to integrate, simply requiring the application of the power rule, which states \(\int x^n \, dx = \frac{x^{n+1}}{n+1}\). Applied here, \(\int 2x \, dx = x^2\).
For the second term, \(\frac{1}{x^2}\), it is helpful to rewrite it as \(x^{-2}\) to apply the power rule again, resulting in \(\int x^{-2} \, dx = -x^{-1} = -\frac{1}{x}\). Breakdowns like these make each step more comprehensible.
Some handy integration techniques to master include:

• Power Rule - for polynomial terms.
• Substitution Method - for more complex integrals that can be simplified through variable changes.
• Integration by Parts - applying to products of functions where substitution is too complex.
• Partial Fractions - generally used for rational expressions.

By practicing these techniques, you can strengthen your ability to tackle integrals with confidence.

Area Under a Curve

The concept of finding the area under a curve is central to understanding definite integrals in calculus. This technique is not only vital in theoretical mathematics but also in real-world applications such as physics and engineering.
The area under a curve represents the accumulation of quantities, and in this particular instance, it translates to calculating the total area beneath the function \(y=2x + \frac{1}{x^2}\) between the bounds of 1 and 4.
By using definite integration, we're essentially summing up these infinitesimally small slices of areas (rectangles under the curve) over the specified interval. The integral \(\int_{1}^{4} \left(2x + \frac{1}{x^2}\right) \, dx\) helps compute this total area efficiently.
This calculation involved several steps from setting up the integral, integrating it term by term, to finally substituting the bounds in to find the net area.
Understanding this process is vital as determining areas under curves helps:

• Calculate distances traveled, when velocity over time is concerned.
• Evaluate economic profits where cost and revenue functions are modeled mathematically.
• Find probabilities in statistics by integrating probability density functions.

Mastering this concept is crucial for solving numerous practical and theoretical problems.