Problem 11 - Calculus an Applied Approach

Question

Find the function values. $$ f(x, y)=\int_{x}^{y}(2 t-3) d t $$ $$ \begin{array}{ll}{\text { (a) } f(1,2)} & {\text { (b) } f(1,4)}\end{array} $$

Step-by-Step Solution

Answer

The values of the function are \( f(1,2) = 0 \) and \( f(1,4) = 6 \).

Step 1: Compute the Antiderivative

To find the antiderivative, divide the exponent of \(t\) in the integral by the new power and subtract the constant. The antiderivative of the integral \( \int (2t - 3) dt \) is \( t^2 - 3t \).

Step 2: Evaluate f(1,2)

Substitute the limits of integration into the antiderivative for this input pair. The result will look something like this: \( [(2)^2 - 3(2)] - [(1)^2 - 3(1)] = 2 - 2 = 0. \) So, \( f(1,2) = 0 \)

Step 3: Evaluate f(1,4)

Do the same for the second input pair. It should be evaluated as follows: \( [(4)^2 - 3(4)] - [(1)^2 - 3(1)] = 8 - 2 = 6. \) So, \( f(1,4) = 6 \).

Key Concepts

Understanding the Antiderivative

When working with definite integrals, finding the antiderivative or the indefinite integral of a function is the first step. An antiderivative is a function whose derivative gives back the original function. For the given exercise, we have the integrand \(2t - 3\).
To find the antiderivative, we perform a reverse process of differentiation. This means integrating \(2t - 3\) with respect to \(t\). The rule of power integration tells us to increase the power of \(t\) by one and divide the coefficient by this power. Hence:

• The antiderivative of \(2t\) is \(t^2\), as we add 1 to the exponent of \(t\) and then divide the coefficient 2 by this new exponent.
• The antiderivative of a constant \(-3\) is \(-3t\) because the integral of a constant \(c\) is \(ct\).

Thus, the antiderivative of \(2t - 3\) is \(t^2 - 3t\). This fundamental step sets the stage for evaluating the integrals.

Calculating Function Values with Definite Integrals

In the context of definite integrals, the function values are found by substituting the boundaries into the antiderivative.
Each pair \((x,y)\) in the problem represents the limits of integration. Once you have the antiderivative \(t^2 - 3t\), you replace \(t\) with the upper limit \(y\) and lower limit \(x\), then subtract:

• For \(f(1,2)\), you plug in, leading to \( [(2)^2 - 3(2)] - [(1)^2 - 3(1)] = 4 - 6 - (1 - 3) = 0\).
• For \(f(1,4)\), calculate as \([(4)^2 - 3(4)] - [(1)^2 - 3(1)] = 16 - 12 - (1 - 3) = 6\).

These computation steps provide the function values at the specified limits. The subtraction allows for evaluation over an interval, reflecting the accumulated change over that span.

Steps to Evaluate Definite Integrals

Evaluating integrals is a pivotal skill in calculus. It transforms the antiderivative into numerical values based on the specified limits.
Every evaluation of a definite integral can be broken down into clear steps:

• Identify the function you need to integrate, as seen with \(2t - 3\).
• Compute its antiderivative, which here was \(t^2 - 3t\).
• Substitute the upper limit into the antiderivative, then subtract the result of the lower limit substitution.
• Perform the arithmetic simplifications to get the final answer.

This systematic approach ensures accurate results and helps in understanding the area or total change represented by the integral.