Problem 44 - Calculus and its applications

Question

Evaluate. $$ \int_{1}^{2}\left(4 t^{3}-1\right) d t $$

Step-by-Step Solution

Answer

The value of the definite integral is 14.

Step 1: Identify the Function to Integrate

The function to be integrated is \( 4t^3 - 1 \). This is a polynomial function of \( t \).

Step 2: Set Up the Integral

The definite integral is set up over the interval from 1 to 2. The integral to solve is \[ \int_{1}^{2} (4t^3 - 1) \, dt. \]

Step 3: Integrate the Function

Find the antiderivative by integrating the polynomial function. The integral of \( 4t^3 \) is \( t^4 \) and the integral of \( -1 \) is \( -t \). So, the antiderivative is \[ t^4 - t + C. \]

Step 4: Evaluate the Antiderivative at Upper and Lower Limits

Substitute the upper limit (2) and the lower limit (1) into the antiderivative. First, evaluate at 2:\[ (2)^4 - 2 = 16 - 2 = 14. \]Then evaluate at 1:\[ (1)^4 - 1 = 1 - 1 = 0. \]

Step 5: Calculate the Definite Integral

Subtract the value of the antiderivative at the lower limit from the value at the upper limit: \[ 14 - 0 = 14. \]

Key Concepts

Understanding Antiderivatives

In calculus, an antiderivative is a function whose derivative matches the original function you started with. It's like working backwards in calculus. When you are finding an antiderivative, often called finding an indefinite integral, you are essentially asking, "What function could I differentiate to get this result?"

For the given problem, we want to find the antiderivative of the polynomial function \(4t^3 - 1\).

• Starting with the function \(4t^3\): Its antiderivative is \(t^4\), because if you differentiate \(t^4\), you get \(4t^3\).
• For the constant term \(-1\), the antiderivative is \(-t\), since the derivative of \(-t\) is \(-1\).
• Thus, the antiderivative of \(4t^3 - 1\) is \(t^4 - t + C\), where \(C\) is the constant of integration, not needed when computing definite integrals.

So, finding antiderivatives involves reversing differentiation to reconstruct the original function, plus a constant!

Exploring Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of a variable, each multiplied by a coefficient. They can be as simple as a constant or as complex as a polynomial with many terms.

In the context of the problem, the function \(4t^3 - 1\) is a cubic polynomial:

• The term \(4t^3\) indicates its degree is 3, the highest power of the variable \(t\). The degree tells you about the polynomial's general shape and behavior - cubic polynomials can have either one or two turning points.
• The constant term \(-1\) shifts the entire polynomial up or down on the graph, but does not affect the overall shape of the graph itself.

Polynomials are important in calculus because they allow us to deal with complex curves and movements in a structured way. Despite their complexity, they offer mathematical simplicity for differentiation and integration. Therefore, understanding polynomial functions is key to mastering integral calculus.

Integration Demystified

Integration is a core concept in calculus. It represents the process of finding the integral of a function, which can either be a definite or an indefinite integral.

• Indefinite Integrals: Often called antiderivatives, they seek to reverse the differentiation process. The result is a function plus a constant \(C\).
• Definite Integrals: They compute the area under a curve within a specific interval. In this case, the definite integral of \(4t^3 - 1\) from 1 to 2 calculates the net area between the function and the \(t\)-axis from \(t = 1\) to \(t = 2\).

To solve a definite integral, we use the "Fundamental Theorem of Calculus," which involves evaluating the antiderivative at the upper limit and subtracting the value at the lower limit.
Integration is incredibly useful in real-world applications such as calculating areas, volumes, central points, and many more physical quantities. It bridges the gap between geometry and algebra, offering insights into a continuous accumulation of quantities.